Page 337 - Rock Mechanics For Underground Mining

ROCK–SUPPORT INTERACTION ANALYSIS

Assume that within the fractured zone

ε 3 = ε 3e − f (ε 1 − ε 1e ) (11.4)

where ε 1e , ε 3e are the strains at the elastic-plastic boundary and f is an experimentally
determined constant as defined in Figure 11.5. Substitution of
(p − p 1 )
ε 1e =−ε 3e = (11.5)
2G
into equation 11.4 and rearranging gives
(p − p 1 )
ε 3 =− f ε 1 − (1 − f ) (11.6)
2G
From equations 11.2, 11.3 and 11.6
du u (p − p 1 )
=− f + (1 − f )
dr r 2G
The solution to this differential equation is
(1 − f )(p − p 1 )

− f
u = Cr + r
2G(1 + f )
where C is a constant of integration which may be evaluated by substituting the value
of ε 1 at r = r e given by equation 11.5. This leads to the solution
u (p − p 1 ) ( f − 1) r e 1+ f
=− + (11.7)
r G(1 + f ) 2 r
Equation 11.7 can be used to plot a relation between radial displacement, generally
represented by i =−u i , and support pressure, p i , at the excavation periphery where
r = a. The differences between the displacements experienced by the rock in the roof,
sidewalls and floor can be estimated by assuming that, in the floor, the resultant support
pressure is the applied pressure, p i , less a pressure that is equivalent to the weight of
the rock in the fractured zone, (r e − a). In the sidewall, the support pressure is p,
and in the roof, gravity acts on the fractured zone to increase the resultant support
pressure to p i + (r e − a).
Consider as an example, a circular tunnel of radius a = 3 m excavated in a rock
mass subjected to a hydrostatic stress field of p = 10 MPa. The properties of the rock
−3 f
◦
◦
mass are = 25 kN m , G = 600 MPa, f = 2.0, = 45 , = 30 and C =
2.414 MPa, which give the parameter values b = 5.828, C 0 = 11.657 and d = 3.0.
An internal radial support pressure of p i = 0.2 MPa is applied.
From equation 7.16, the radius of the fractured zone is calculated as
1/(d−1)

2p − C 0
r e = a = 7.415 m
(1 + b)p i
and the radial pressure at the interface between the elastic and fractured zones is given
by equation 7.14 as p 1 = 1.222 MPa. The radial displacement at the tunnel periphery
is then given by equation 11.7 as

i =−u i = 0.228 m
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