Page 256 - Rock Mechanics For Underground Mining

EXCAVATION DESIGN IN STRATIFIED ROCK

Combining equations 8.17 and 8.26

1/2 3
s

2
n ε xx 3 2 ε xx 1 + 16 z
z = = 1 − 1 − 1 + s z = (8.27)
z on r 16 2r 1 − z
2
8.5.4 Stress-strain relations
In the elastic range of beam deformation, the stress-strain relations are
xx = ε xx E, m = ε m E (8.28)

so that, from equation 8.27,
2

z n
Er 1 −
z on
xx = 3 (8.29)
1 + s 2
16 z
Substituting for xx in equation 8.14, the equilibrium condition expressed in terms of
strain is
1 s 2 s z q n s n 2
ε xx = = Q n = (8.30)
4 Enz 4n(1 − z ) [4nz on (1 − z )]
where
s s s n n
Q n = q n s n = , s z = = , z = =
E z 0 z on z 0 z on
Introducing the expression for arch strain from equation 8.30 in the kinematic com-
patibility equation 8.27 yields the arch deformation equation:
3 2

1 + s
16 z
z = Q n s z
z
8nr 1 − (1 − z )
2
3 2

1 + s
2 16 z
z (8.31)
= q n s z on z
8nr 1 − (1 − z )
2
With n and r defined by equations 8.15 and 8.25, equations 8.31 and 8.29 provide the
means of determining the deflection, abutment stress and stability of the beam.
8.5.5 Resistance to buckling or snap-through
Elastic stability of the voussoir beam is assured if the maximum possible resisting
moment M R is greater than the disturbing moment M A . A necessary condition is that
the resisting moment increases with increasing deflection. From equations 8.17 and
8.19,
dM R dM R
=− ≥ 0 (8.32)
d n dz n
1
dM R 2 z n d xx
= nt + xx (8.33)
dz n 2 dz n
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