Foundations: stress distributions beneath variably loaded areas 329influence function may be developed and used through implementation
of the sector method. The principle is that the uniformly loaded area is
divided into sectors around the point of interest, analytical integration
performed over each sector and the total effect found by summation of
the sectorial contributions. The technique can be conducted in graphical,
semi-graphical or numerical fashion. Figure 18.14 demonstrates the basic
principle.
A loaded area with an irregular boundary is shown in Fig. 18.14(a).
Around some arbitrarily chosen point of interest a number of sectors have
been drawn. Figure 18.14@) shows a typical sector in detail, indicating
an element over which the analytical integration will be performed. The
subtended angle at the origin of the sector is assumed to be sufficiently
small to enable adequate representation of the irregularity of the
boundary.
Considering the element shown in Fig. 18.14@),
elemental load = p b 68 &.
As an example, consider the formula for the vertical displacement due to
a normal point load pen in Fig. 18.12, i.e.
We then substitute the elemental load for the point load P at the elemental
position z = 0 and R = b. This reduces to
To obtain the displacement induced by loading over the complete sector,
the above expression is integrated for b = 0 to b = r, giving
and finally, for the total loaded area,
Evaluation of the term Zr68 involves either graphical, semi-graphical or
numerical techniques to determine a value of r for each value of 8. In most
cases, the number of sectors required to produce a result of acceptable
accuracy is modest-as the reader can verify for the case of a circular
area using the formula above, knowing that the analytical result to
Cr68 is 2m.
The sector method is a simplified version of the stress influence function
method, where the loading is uniform over the entire area and polar co-
ordinates have been used. Given the conditions of a uniform load, the