Geotechnical Engineering

DHARM

354 GEOTECHNICAL ENGINEERING

shear stress acts. In Fig. 10.1 (c), the cylindrical co-ordinates and the corresponding normal
stresses—radial stress σr, tangential stress σt, and the shear stress τrz—are shown; σz is an-
other principal stress in the cylindrical co-ordinates; the polar radial stress σR is also shown.

R= x+y+z

Ö

222

r= x +y

Ö 22

y

sy

z

sx A

Y Z¢

Z¢

Q

O

Y¢

x

X¢ X

q

(a) (b) (c)

sz

tzx

tzy

sx

tyz txy

tyx

sy

Z

Q

sR

trz

sr

st

Z¢

sz

txz

q

Fig. 10.1 Notation for Boussinesq’s analysis

The Boussinesq equations are as follows:

σz =

3

2

3

5

Qz

π R

. ...(Eq. 10.2 (a))

=

3

2

2

2

Q

π z

θ

.

cos

...(Eq. 10.2 (b))

=

3

2

3

2252

Qz

π rz

.

()+ /

...(Eq. 10.2 (c))

=

3

2

1

(^221)
52
Q
π+zrz
L
N
M
M
O
Q
P
(/ )P
/
...(Eq. 10.2 (d))
σx = Qxz
R
xy
Rr R z
yz
(^2) Rr
(^3212)
5
22
2
2
π −−υ^32
−

• 
  


• 
  

  R
  S
  T
  U
  V
  W
  L
  N
  M
  M
  O
  Q
  P
  P
  ()
  ()
  ...(Eq. 10.3)
  σy =
  Qyz
  R
  yx
  Rr R z
  xz
  (^2) Rr
  (^3212)
  5
  22
  2
  2
  π −−υ^32
  −

  
  


• 
  


• 
  

  R
  S
  T
  U
  V
  W
  L
  N
  M
  M
  O
  Q
  P
  P
  ()
  ()
  ...(Eq. 10.4)
  σR =
  3
  2 2
  Q
  π R
  .cosθ
  ...(Eq. 10.5)
  σr =
  Qzr
  (^2) R RR z
  3122
  π^2
  − − υ

  
  


• 
  

  L
  N
  M
  O
  Q
  P
  ()
  ()
  ...(Eq. 10.6 (a))