Geotechnical Engineering

DHARM

358 GEOTECHNICAL ENGINEERING

Vertical Stress Distribution Along a Vertical Line

The variation of vertical stress with depth at a constant radial distance from the axis of
the load may be shown by horizontal ordinates as in Fig. 10.4.

Q r

q»

¢

²

39°13 53.5

+Z

szmax

Fig. 10.4 Vertical stress distribution along a vertical line at radial distance r
As z increases, r/z decreases for a constant value of r. As r/z decreases KB-value in the
equation for σz increases, but since z^2 is involved in the denominator of the expression for σz,
its value first increases with depth, attains a maximum value, and then decreases with further
increase in depth. It can be shown that the maximum value of σz occurs when the angle θ made

by the polar ray attains a value 39°13′53.5′′, corresponding to a value of 23/ or 0.817 for r/z;

the maximum value σz is then 0.0888 Q. This value decreases rapidly with depth; for r/z = 0.1,
the value is just 0.0047 Q.

The values are tabulated for convenience as shown below:

Table 10.2 Variation of vertical stress with depth at

constant value of r (Say r = 1 unit)

Depth z (Units) r/z KB KB/z^2 σz

0 ∞ – – – – Indeterminate

0.5 2.0 0.0085 0.0340 0.0340 Q

1 1.0 0.0844 0.0844 0.0844 Q

2 0.5 0.2733 0.0683 0.0683 Q

5 0.2 0.4329 0.0173 0.0173 Q

10 0.1 0.4657 0.0047 0.0047 Q