Page 255 - Rock Mechanics For Underground Mining

ROOF BEAM ANALYSIS FOR LARGE VERTICAL DEFLECTION

arch (the thrust line) operating longitudinally in the beam. Suppose the arch profile
is described by the general parabolic expression

2
s = 4az

where s is the beam half span and a is chosen to satisfy the assumed arch geometry.
Elementary analytic geometry and the theory of hyperbolic functions then allow the
arc length, L, of the arch to be expressed in terms of its height, z, and span, s,by

16 z 2
L = s + (8.20)
3 s
It is noted that this expression for arc length differs from that used by Diederichs and
Kaiser (1999a).
For the undeflected arch,
16 z 2 0
L 0 = s + (8.21)
3 s
Introducing the mean state of axial strain in the beam, ε m , the shortening of the arch
and the moment arm z after deflection are given by

L = L o − L = ε m L o (8.22)
1/2

3 s
2
z = z − (8.23)
0
16 L
8.5.3 Relation between deflection and strain
The mean strain in the arch, ε m , is related to the outer fibre axial strain, ε xx (which
corresponds to the outer fibre stress xx ), by the expression
ε xx
r = = 2 (8.24)
ε m n
where corresponds to the mean depth of the arch at the quarter span.
The parameter r is equivalent to a strain concentration factor arising from the
triangular distribution of normal stress at the abutments and central crack. From an
analysis of results of finite element studies, Sofianos (1996) showed that r may be
estimated from the expression

1
r = n ⇔ = 0.19 (0.1 ≤ n ≤ 0.3) (8.25)
2.6
Equations 8.22, 8.23 and 8.24 may be combined to yield
1/2

ε xx 3 2
z n = z on 1 − 1 + s z (8.26)
r 16
where the subscript z denotes normalisation of geometric parameters with respect to
the initial moment arm z o .
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