Page 346 - Rock Mechanics For Underground Mining

ROCK SUPPORT AND REINFORCEMENT

If, for example, T = 10 tonne = 100 kN, = 25 kN m −3 and D = 4 m, equation
11.8 gives s = 1.0m.
In this application, care must be taken to ensure that the bolt anchors have an
adequate factor of safety against failure under the working load, T . This design
method is conservative in that it does not allow for the shear or flexural strength of
the strata above the abutments.
Lang and Bischoff (1982) extended this elementary analysis to incorporate the
shear strength developed by the rock mass on the vertical boundaries of the rock unit
reinforced by a single rockbolt. The rock is assumed to be destressed to a depth, D,
as in Figure 11.13, but variable vertical stresses, v , and horizontal stresses, k v , are
assumed to be induced within the de-stressed zone. Typically, k may be taken as 0.5.
The shear strength developed at any point on the perimeter of the reinforced rock
unit is given by c +
k v , where c is the cohesion and
= tan is the coefficient of
friction for the rock mass. Lang and Bischoff’s analysis leads to the result
T c
1 − exp(−
kD/R)
= 1 − (11.9)
AR
k R 1 − exp(−
kL/R)
2
where T = rockbolt tension, A = area of roof carrying one bolt (= s for a s × s
bolt spacing), R = shear radius of the reinforced rock unit, = A/P, where P is the
shear perimeter (= 4s for a s × s bolt spacing), = a factor depending on the time
of installation of the rockbolts ( = 0.5 for active support, and = 1.0 for passive
reinforcement), and L = bolt length which will often be less than D, the height of
the de-stressed zone of rock.
Lang and Bischoff suggest that, for preliminary analyses, the cohesion, c, should
be taken as zero. Design charts based on equation 11.9 show that, particularly for
low values of , the required bolt tension, T , increases significantly as L/s decreases
below about two, but that no significant reduction in T is produced when L/s is
increased above two. This result provides some corroboration of Lang’s empirical
rule that the bolt length should be at least twice the spacing. For a given set of data,
equation 11.9 will give a lower required bolt tension than that given by equation
11.8. Clearly, Lang and Bischoff’s theory applies more directly to the case of the
development of a zone of reinforced, self-supporting rock, than to the simpler case
of the support of the total gravity load produced by a loosened volume of rock or by
a roof beam in laminated rock.

Design to support a triangular or tetrahedral block. InChapter9,theidentification
of potential failure modes of triangular and tetrahedral blocks was discussed, and
analyses were proposed for the cases of symmetric and asymmetric triangular roof
prisms. These analyses take account of induced elastic stresses and discontinuity
deformability, as well as allowing for the self-weight of the block and for support
forces. The complete analysis of a non-regular tetrahedral wedge is more complex.
An otherwise complete solution for the tetrahedral wedge which does not allow for
induced elastic stresses is given by Hoek and Brown (1980).
The analyses presented in Chapter 9 may be incorporated into the design proce-
dure. Consider the two-dimensional problem illustrated in Figure 11.14 to which the
analysis for an asymmetric triangular prism may be applied. If it is assumed that the
normal stiffnesses of both discontinuities are much greater than the shear stiffnesses,
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