Engineering Rock Mechanics

Foundations: stress distributions beneath variably loaded areas 327

Uniform pressure applied to a rectangular base

0

r '9 Rectangular area

in the xy plane

Point of interest subjected to

on z-axis at

depth z below

xy plane

uniform normal

pressure p

Figure 18.13 Integration of Boussinesq and Cerruti solutions over each component
element of a loaded area.

determining influence functions for the component areas. This technique
applies, with suitable variation, to all of the components of stress and
displacement.
In Figure 18.13, the area bounded by XI, X,, Y1 and Yz is assumed to be
loaded with a uniform normal stress, p, and we wish to consider the
consequential stress component o, and displacement component u, at the
point F at depth z below the surface of the half-space. These are found by
integrating the relevant expressions given in Fig. 18.22 over the loaded area.
Considering a small element dx - dy, as shown in Fig. 18.13, the equivalent
point load is P = pax x Sy and thus the relevant expression for the stress
component induced by this infinitesimal element is:

oz = gdxdy.

2n~~

To calculate the total stress component at F, integrate between the
appropriate limits in the x-y plane as follows:

Although z is independent of x and y, R = (x2 + 4 + z'), with the result
that evaluating the integral is not straightforward. However, a standard
form exists for the integral (ref. Handbook of Mathematical Functions,
Abromovitch and Stegun, 1965) as

The term I,(x,y) is referred to as a stress influencefunction and the stress itself
is therefore given as