Page 263 - Rock Mechanics For Underground Mining

IDENTIFICATION OF POTENTIAL BLOCK FAILURE MODES – BLOCK THEORY

In the two-dimensional analysis, the prisms EP and JP are represented by plane
angles with a common apex O. In three dimensions, the prisms are pyramids with a
common vertex O.

9.2.2 Stereographic analysis
The elucidation of block geometry requires the use of a full stereographic projection.
The projection has the advantage of allowing a fully three-dimensional problem to
be analysed in two dimensions. The way in which the full projection is constructed is
illustrated in Figure 9.4. Considering an upper hemisphere projection, Figure 9.4(a)
illustrates a vertical section through a reference sphere of radius R. Using the point F at
the base of the sphere as the focus for the projection, a point A on the upper hemisphere
(representing the line OA) has a projection OA 0 on the extended horizontal diametral
line in Figure 9.4(a), which represents the equatorial plane of the reference sphere.
The location of A 0 is given by

OA 0 = R tan /2 (9.1)

where is the angle between the vertical axis and the radius OA.
For a point B on the lower hemisphere defined by the angle , its projection B 0 on
the equatorial line is given by the distance OB 0 ; i.e. B 0 plots outside the horizontal
diameter of the projection circle:

OB 0 = R cot /2 (9.2)

The complete upper hemisphere stereographic projection of two orthogonal sets of
uniformly spaced planes, interesting the reference sphere as great circles and small
circles, is shown in Figure 9.4(b).
◦
Consider the complete stereographic projection of a plane oriented 30 /90 (dip/dip
◦
direction) and its two half-spaces, as shown in Figure 9.5. In the projection, the region
between the part of the plane’s great circle that falls inside the reference circle and the
boundary of the reference circle represents all the lines that are directed through the
◦
centre of the reference sphere into the half-space above plane 30 /90 . Similarly
◦
the region bounded by the reference circle and the part of the plane’s great circle that
lies outside the reference circle represents all the lines that are directed into the lower
◦
◦
◦
◦
half-space of plane 30 /90 . If the great circle of the plane 30 /90 represents joint
set 1, then the region inside the reference circle represents its upper half-space, i.e.
U 1 , and the region outside it represents the lower half-space L 1 .
The Joint Pyramid (JP). Consider the three joint sets shown as great circles in
Figure 9.6. The intersections of the three great circles yield eight spherical triangles
(each identified by three lines of intersection of the great circles), each of which
corresponds to a trihedral angle with its vertex at the centre of the reference sphere.
For the point marked A, the segments of the spherical triangle surrounding it represent
the upper half spaces of planes 1, 2 and 3. Introducing the notation where digits 0
and 1 represent the upper and lower half-spaces of a joint and the digits are ordered
according to the order of numbering of the joint sets, the point A is surrounded by
a triangle formed of sides which represent the upper half spaces of the planes 1, 2
and 3, and is denoted 000. On the other hand, point B lies inside the great circle for
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