Engineering Rock Mechanics

Questions and answers: rock reinforcement and rock support 2 79

25r-- -

-/ --

__

operating point:

u=5.9mm, p7.3MPa .- __

I

0 2 4 6 8 10 12 14 16

radial displacement, mm

-1 I- 1 mm displacement of tunnel boundary

This shows how, by delaying the installation of the lining, we have

reduced the pressure it is required to withstand, but at the expense of

increasing the final radial displacement.

416.70 The sketch shows a pillar of rectangular cross section, which

is very long in the y-direction (i.e. out of the plane of the paper)

and is made of a linearly elastic isotropic rock. A support pressure

is applied to the horizontal sides of the pillar through the action of

springs that pass horizontally through the pillar.

(a) Develop an expression linking the vertical stress, cr, to the

support pressure, p, in terms of the elastic constants of the pillar

material and the spring stiffness. Use this expression to show that,

for zero horizontal strain of the pillar, the ratio of vertical stress to

support pressure is (1 - u)/w.

(b) If the strength of the pillar can be described by the Hoek-

Brown criterion with uc = 37 MPa, s = 1 and m = 15, and the

elastic modulus and Poisson's ratio of the rock are 75 GPa and

0.27, respectively, what spring stiffness is required if the pillar is to

withstand a vertical stress of 55 MPa?

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A 7 6.7 0 (a) The generalized Hooke law for an isotropic material can be
written directly as both

E, = - [ox - u (ay + az)] and E - - [ay - u (az + a,)].

Because the pillar is very long in the y-direction, we can say that it is

in a state of plane strain and hence sy = 0. Substituting this and a, = p

in the second of these equations gives

and then substituting this in the first equation leads to

1 1

E y- E

cy = v(az + P)

1

E

E, = - [p (1 - 2) - azv (1 + u)].