Engineering Rock Mechanics

328 Design and analysis of surface excavations

Once all the influence functions have been evaluated (these being
geometrical functions), the stresses and displacements at any point can be
found as the result of any loading distribution-by discretizing the surface
loading appropriately and applying the relevant influence functions to
summate the individual contributions made by each element.
Exactly the same approach is used for the displacements. The total
vertical displacement induced by the element 6x - Sy is given by

from which the displacement influence function is evaluated as

Again, the total displacement induced by the loading over the particular

element is calculated as

As with the stresses, the displacement contributions from each of the

individual uniformly loaded discrete component elements are added to

give the total displacement at the point F.

78.3.3 The sector method

In the circumstances of an irregular boundary of a uniformly loaded area,

analytical integration of the Boussinesq and Cerruti solutions may be either

intractable or impossible, but a simplified form of the stress or displacement

Typical sector

Uniform pressure p

over irregular area

tY

Elemental

load = pbEb

/

Area of

element

(b)

Figure 18.14 The sector method for loaded irregularly-shaped areas: (a) irregularly-

shaped areas are divided into sectors; and @) geometry of a typical sector.