Engineering Rock Mechanics

Statistical analysis of stress state data 53

stress components. Thus, the variability (expressed via the six standard
deviations of the components) is in itself a tensor with its own principal
values and directions-which may not coincide with the mean principal
stress directions. The subject of tensor statistics is, however, beyond the
scope of this book.
The point being made is that the correct procedure for determining mean
stresses must be utilized. Also, the form of the variability of several
measurements made in one location can in itself be diagnostic. The direct
procedure for establishing mean principal stresses from two stress tensors,
say, the results of two stress determinations, is shown in the box below. The
method outlined in the box can, of course, be extrapolated to any number
of tensors.

Correct method for averaging two stress

tensors

Two principal stress tensors resulting from stress measurement prog-

rammes are shown below and identified by the superscripts u and b:

The principal stress components in these tensors will generally have

different orientations. Before averaging can proceed, these must be

transformed to a common set of reference axes, thus:

When averaged, these tensors give a subsequent tensor,

from which can be calculated the 'global' average principal stress tensor:

together with the directions of the principal stresses.