DHARM

STRESS DISTRIBUTION IN SOIL 355

=

Qzr

(^2) rz rzzrz
312 2
π^22522222
υ
()
()

• / −
  −
  ++ +
  L
  N
  M
  M
  O
  Q
  P
  P
  ...(Eq. 10.6 (b))
  = Q
  2 z
  3 12
  (^21)
  23
  2
  π
  θθ υθ
  θ
  sin cos ()cos
  ( cos )
  − −


• 
  

  L
  N
  M
  O
  Q
  P ...(Eq. 10.6 (c))
  σt = −− −

  
  


• 
  

  L
  N
  M
  O
  Q
  P
  Q
  2 z
  12
  (^21)
  3 2
  π
  υθ
  θ
  θ
  ()cos
  cos
  cos
  ...(Eq. 10.7 (a))

  
  

  −
  −

  
  


• 
  

  −
  ++ +
  L
  N
  M
  M
  O
  Q
  P
  P
  Qz
  (^2) rz rzzrz
  12
  1
  π()υ ()^2232 /^2222 ...(Eq. 10.7 (b))
  τrz =
  3
  2
  2
  5
  Qrz
  π R

  
  

. ...(Eq. 10.8 (a))

=

3

2

1

(^321)
52
Qr
π+zrz
L
N
M
M
O
Q
P
(/ )P
/
...(Eq. 10.8 (b))
= (3Q/2πz^2 ). (sin θ cos^4 θ) ...(Eq. 10.8 (c))
Here υ is ‘Poisson’s ratio’ of the soil medium.
A geotechnical engineer must understand the assumptions on which these formulae are
based, in order to be able to identify those problems to which they are directly applicable and
those in which some modifications are necessary. There is usually no need for one to under-
stand the advanced mathematical procedures by which the solution was obtained. For proofs,
the reader is referred to Timoshenko and Goodier (1951) and Jumikis (1962).
Some modern methods of settlement analysis, such as those proposed by Lambe (1964,
1967), necessitate determining the increments of both major and minor principal stresses;
however, in most foundation problems it is only necessary to be acquainted with the increase
in vertical stresses (for settlement analysis) and the increase in shear stresses (for shear strength
analysis).
Equation 10.2 (d) may be rewritten in the form:
σz = K
Q
Bz

. 2 ...(Eq. 10.9)

where KB, Boussinesq’s influence factor, is given by:

KB =

(/ )

[(/)]/

32

1 252

π

+ rz

...(Eq. 10.10)

The influence factor is a function of r/z as shown in Fig. 10.2.
Gilboy (1933) has prepared a table of Boussinesq’s influence coefficients for a large range
of values of r/z. (KB is as low as 0.0001 for r/z value 6.15). It is interesting to note that the
influence factors for shearing stress, τrz, can be found by multiplying the KB-values for σz by
the r/z-ratio. The intensity of vertical stress, directly below the point load, on its axis of load-
ing, is given by:

σz =

0 4775

2

. Q
z

...(Eq. 10.11)