Page 474 - Rock Mechanics For Underground Mining

LONGWALL AND CAVING MINING METHODS

value of the semi-minor axis of the ellipsoid can be calculated as

1/2
E N
b N = (15.6)
2.094 h N
or as
h N 2 1/2
b N = (1 − ε ) (15.7)
2
For a given ellipsoid of motion of volume E N , there will be a corresponding limit
ellipsoid of volume E G , beyond which the material remains stationary. The material
contained between the boundaries of the two ellipsoids will loosen and displace,
but will not report to the discharge point. Janelid and Kvapil (1966) described this
loosening by a factor

E G
=
E G − E N
They found that varies between 1.066 and 1.100. For most broken ores, tends
towards the lower end of this range which gives

E G 15E N (15.8)

Assuming that the limit ellipsoid has the same eccentricity as the ellipsoid of motion,
equations 15.6, 15.7 and 15.8 can be used to calculate its height as

(15.9)
h G 2.5h N
As material is progressively discharged, the size of the ellipsoid of motion, and
of the corresponding limit ellipsoid, continues to grow. A dimension required in the
design of sublevel caving layouts, is the radius of the limit ellipsoid at the height h N
(Figure 15.21)
2
r [h N (h G − h N )(1 − ε )] 1/2 (15.10)

The analysis so far assumes that flow is symmetric about a vertical axis. In sublevel
caving, the boundary conditions are often such that the ellipsoid of motion and the
corresponding limit ellipsoid are not fully developed. In a vertical section through the
longitudinal axis of a production heading in the general case, the ellipsoid of motion
is truncated by the wall of the unblasted ore (Figure 15.23). In addition, the centre
line of the ellipsoid is deviated away from the wall by an angle, , which varies with
the ring gradient, , and with the angle of friction developed between the broken and
unbroken ore.
The “theoretical” flow pattern in the plane of the wall of a vertical slice is shown
in Figure 15.24. The shapes of the truncated ellipsoid of motion and limit ellipsoid
illustrated in Figure 15.23 vary with the particle size of the broken ore, the height
of the flow, the extraction width and the extraction velocity. The flow becomes more
narrow, or more parallel, as the height of the flow zone increases, as the extraction
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