Civil Engineering Reference
In-Depth Information
(16.5)
This linear equation system has an infi nite number of solutions which can be expressed as
(16.6)
where
is an arbitrary real number and {ei}
i
} is a unit vector. The components of {ei}
i
}
represent the direction cosine of
λ
σ
i
with respect to the coordinate system (x,y,z):
(16.7)
in which vi
i
is the absolute value of {vi}
i
} and (i,x), (i,y), (i,z) are the angles between
σ
i
and
the coordinates x, y and z. These nine angles determine the orientations of the principal
normal stresses in three dimensions.
Figure 16.6 shows the result of a stress measurement in claystone using the CSIRO HI cell.
In Fig. 16.6 (left) the magnitudes and orientations of the principal normal stresses are illus-
trated by means of an isometric plot. The principal normal stresses can also be represented
as points in a lower hemisphere stereographic projection (Fig. 16.6, right). In the latter case
the inclination and direction of the principal normal stresses, that is, the angles of dip and
dip direction, are referred to as “plunge” and “trend”, respectively (ISRM 2003).
Figure 16.6
Representation of orientations and magnitudes of principal normal stresses
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