Engineering Rock Mechanics

2 78 Rock reinforcement and rock support

at a radial stress of 20 MPa, the other is the elastic displacement
induced when the radial stress is zero. We use the Kirsch solution to
find this radial displacement and, for a hydrostatic stress state, it is
given by

ur = -- Pa
2G
where p is the value of the hydrostatic stress, a is the radius of the
tunnel, and G is the shear modulus.
However, as we have not been given a value for the shear modulus,
let us assume a value of 2 GPa. This gives a radial displacement of

pa 20x 1.85

2G 2x2000

u r- ---=- = -0.0925 m.

If we assume that the lining behaves as a thick-walled cylinder subject

to radial loading, then the equation for the lining characteristic is

pr = k- ur - uo

a

where pr is the radial support pressure, k is the lining stiffness, and uo is
the magnitude of the rock displacement when the lining is installed, and

(16.1)

where t is the thickness and the subscript c refers to the concrete lining.
To use Eq. (16.1) we assume values for E, and uc as 30 GPa and
0.25, respectively. Using these figures, together with a = 1.85 m and
tc = 1.85 - 1.70 = 0.15 m, we find that k = 2.78 GPa. Thus, for a radial
pressure of 20 MPa and uo = 0 mm, the lining will deflect radially by
a 1.85
u - -pr + uo = 20 + 0 = 0.013 m.
r- k 2.78 x 103
The operating point can now be found:

0 2 4 6 8 10 12 14 16

radial displacement, mm

(b) If the lining is installed after a radial displacement of 1 mm has

occurred at the tunnel boundary, then we have the following: