Geotechnical Engineering

DHARM

STRESS DISTRIBUTION IN SOIL 373

The principle of superposition may be conveniently employed to compute the stress
beneath any point either inside or outside a uniformly loaded rectangular area. This is illus-
trated as follows:
Let the point A at which the vertical stress is required be at a depth z beneath A′, inside
the uniformly loaded rectangular area PQRS as in Fig. 10.18 (a).

Imagine TU and VW parallel to the sides and passing through A′. σz at A is given by:

σz = qIdiσσ σ σIII III IV++ +I I I ...(Eq. 10.48)

where IIσσIII, ..., are the influence factors for the stress at A due to the rectangular areas as I,
II, ..., by the principle of superposition, since A′ happens to be a corner for these areas.
In the point A is beneath A′, outside the uniformly loaded rectangular area PQRS, as in
Fig. 10.18 (b), imagine PQ, SR, PS and QR to be extended such that PWA′T is a rectangle and
U and V are points on its sides, as shown.

SVR

TU

III

IV III

PWQ

T

S

UA¢

R V

P Q W

A¢

PWA T :¢ I SVA T :¢ II

QWA T :¢ IIIRVA U :¢ IV

(a) Point inside the loaded area (b) Point outside the loaded area

Fig. 10.18 Stress at a point other than under a corner of a rectangular areas

Then σz at A is given by:

σz = qIdiσσ σ σIIIIIIIV−− −I I I ...(10.49)

where,IIσσIII, ..., are the influence factors for the stress at A due to the rectangular areas

designated I, II, ..., by the principle of superposition. (Since area IV is deducted twice, its

influence has to be added once).

10.6.2Uniform Load on Rectangular Area based on Westergaard’s Theory

If the soil conditions correspond to those assumed in Westergaard’s theory, that is, the soil

consists of very thin horizontal sheets of infinite rigidity, which prevent the occurrence of any

lateral strain and the vertical stress at a point below a corner of a uniformly loaded rectangu-

lar area then it may be obtained by integration of the stress due to a point load under similar

conditions, and shown to be:

σz = ( /q ) cot.

mn mn

2

12

22

11 12

22

1 1

22

2

π 22

ν

ν

ν

ν

− −

−

F

HG

I

KJ

F +

HG

I

KJ

+

−

−

F

HG

I

KJ

L

N

M

M

M

O

Q

P

P

P

...(Eq. 10.50)