Geotechnical Engineering

DHARM

STRESS DISTRIBUTION IN SOIL 357

Vertical Stress Distribution on a Horizontal Plane

The vertical stress on a horizontal plane at depth z is given by:

σz = K Q

z

B. 2 , z being a specified depth.

For several assumed values of r, r/z is calculated and KB is found for each, the value of σz
is then computed. For r = 0, σz is the maximum of 0.4775 Q/z^2 ; for r = 2z, it is only about 1.8%
of the maximum, and for r = 3z, it is just 0.3% of the maximum. The distribution is as shown in
Fig. 10.3 and Table 10.1.

Q

rrA

z

0.4775 —Q

z^2

21 r/2 r/2 1 2

Fig. 10.3 Vertical stress distribution on a horizontal plane at depth z (Boussinesq’s)
Theoretically, the stress approaches zero at infinity, although practically speaking, it
reaches a negligible value at a short finite distance. The maximum pressure ordinate is rela-
tively high at shallow elevations and it decreases with increasing depth. In other words, the
bell-shaped figure flattens out with increasing depth.

If Q is taken as unity, this diagram becomes what is known as the ‘Influence Diagram’
for the vertical stress at A. With the aid of such a diagram, it is possible to determine the
vertical stress at point A due to the combined effect of a number of concentrated loads at
different radial distances from A, which will be the summation of the products of each of the
loads and the ordinates of this diagram under each load.

Table 10.1 Variation of vertical stress with radial

distance at a specified depth (z = 1 unit, say)

rr/zKB σz

0 0 0.4775 0.4775 Q

0.25 0.25 0.4103 0.4103 Q

0.50 0.50 0.2733 0.2733 Q

0.75 0.75 0.1565 0.1565 Q

1.00 1.00 0.0844 0.0844 Q

1.25 1.25 0.0454 0.0454 Q

1.50 1.50 0.0251 0.0251 Q

1.75 1.75 0.0144 0.0144 Q

2.00 2.00 0.0085 0.0085 Q