Geotechnical Engineering

DHARM

STRESS DISTRIBUTION IN SOIL 385

∴σz = 450(0.1936 + 0.2290 + 0.1368 + 0.1202)
= 305.8 kN/m^2.
Example 10.9: A rectangular foundation 2 m × 3 m, transmits a pressure of 360 kN/m^2 to the
underlying soil. Determine the vertical stress at a point 1 metre vertically below a point lying
outside the loaded area, 1 metre away from a short edge and 0.5 metre away from a long edge.
Use Boussinesq’s theory.
z = 1 m; q = 360 kN/m^2
since the point at which the stress is required is outside the loaded area, rectangles are imag-
ined as shown in Fig. 10.27, so as to make A′ a corner of all the concerned rectangle. With the
notation of Fig. 10.27,

σz = qIdiσσ σ σIIIIIIIV−− −I I I

Area I: m = 2.5/1 = 2.5; n = 4/1 = 4

IσI =^1

4

21

1

2

1

21

1

22

22 22

22

22

1

22

π^2222

mn m n

mn mn

mn

mn

mn m n

mn mn

++

+++

++

++

+

++

++−

L

N

M

M

O

Q

P

P

.tan−

2m

3m

P Q W

U

S V

A¢

0.5

m

R

T

1m

PWA T :

SV

¢ I

AT

QWA U

PWA U

¢:

¢:

¢:

II

III

IV

Fig. 10.27 Stress at a point outside loaded area (Ex. 10.9)

=

1

4

225425 4 1

25 4 1 25 4

25 4 2

25 4 1

225425 4 1

25 4 1 25 4

22

22 22

22

22

1

22

π^2222

×× ++

+++ ×

++

++

+

×× ++

++− ×

L

N

M

M

O

Q

P

P

.. −

..

.

.

.

tan

..

..

= 0.2434

Area II: m = 0.5/1 = 0.5; n = 4/1 = 4

IσII =^1

4

205405 4 1

05 4 1 05 4

0542

05 4 1

205405 4 1

05 4 1 05 4

22

22 22

22

22

1

22

π^2222

×× ++

+++ ×

++

++

+

×× ++

++− ×

L

N

M

M

O

Q

P

P

.. −

..

.

.

.

tan

..

..

= 0.1372

Area III: m = 1/1 = 1; n = 2.5/1 = 2.5

IσIII=^1

4

21251 25 1

1251125

1252

1251

21251 25 1

1251125

22

22 22

22

22

1

22

π 22 22

×× + +

+++×

++

++

+

×× + +

++−×

L

N

M

M

O

Q

P

P

.. −

..

..
.

tan

..

..

= 0.2024