Page 257 - Rock Mechanics For Underground Mining

ROOF BEAM ANALYSIS FOR LARGE VERTICAL DEFLECTION

and from equation 8.29,

z n
d xx 2
z on
(8.34)
3 2
=−2Er
dz n 1 + s
16 z
Introducing equation 8.34 in equation 8.33, and satisfying the inequality condition of
equation 8.32,
2

1 1 − 3 z n
dM R 2 z on
= nt Er 3 < 0 (8.35)
dz n 2 1 + s 2
16 z
The inequality is satisfied if
z on
z on > z n > √ (8.36)
3
Hence, the maximum resisting moment, corresponding to the minimum height of a
stable arch and the maximum vertical deflection of the arch, is given by

z on
z n = min z n = √ (8.37)
3
and from equation 8.16,
z on
max n = z on − min z n = z on − √ = 0.42z on (8.38)
3
From equation 8.27, this value of n corresponds to an outer fibre elastic strain given
by
2 r
3
3 2
ε xx = (8.39)
1 + s
16 z
In some numerical studies using UDEC, Sofianos considered deflection and moment
arm in the arch at the condition of bucking. The comparison shown graphically in
Figure 8.11 indicates a very good correspondence between the deflection and arch
height at buckling calculated using equations 8.37 and 8.38 and from the numerical
analysis, for a wide range of normalised beam spans.
For stability against buckling, the maximum resisting moment must be greater than
the deflecting moment, or from equations 8.19 and 8.13,
1 2 1 2 M R
M R = xx nt min z n ≥ qs = M A = (8.40)
2 8 FS b
where q is the distributed load on the beam.
Substituting in equation 8.40 the expressions for xx and min z n (equations 8.29
and 8.37) yields the following expression governing elastic stability and buckling of
the voussoir arch, for known values of n and r:
3 2

s
√ 1 + 16 z 1
3 3Q n s n = S b ≤ 1 (8.41)
8rn F
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