ABSTRACT
Figure SI 19.1 shows in simple cross-section form the conditions that apply to the analysis of the problem of a pressure tunnel in rock subjected to a uniform state of in situ stress.
The following primary variables are used in the analysis, first of a thick-walled concrete cylinder that is subjected to unrestrained dilation when internal hydrostatic pressure is applied, and, second, when any dilation at the extrados of the concrete cylinder is resisted by contiguous, relaxed and broken (perhaps grouted) rock: r is the radius from centre of circular, concrete-
lined pressure tunnel to any point of interest (in the lining or the rock mass)
rt is the radius to the inside concrete surface ra is the radius to the outside concrete surface (ra - ry) is the concrete thickness a=ra/ri ct is the water pressure in the tunnel a is the external (to the tunnel) pressure (due to
stress field in the rock) 8bi is the displacement of a point on the inner
(water) side of the concrete lining 8ba is the displacement of a point on the outer
(rock contact) side of the concrete lining vb is Poisson's ratio of concrete vg is Poisson's ratio of the rock mass adjacent to
the lining extrados mb is v^-1 (Poisson's number) for the concrete mg is vr'1 (Poisson's number for the rock) Eb is Young's modulus for the concrete
Figure SI 19.1 Assumed pressure conditions, internal and external, in a circular concrete-lined tunnel in a hydrostatic stress field
Eg is Young's modulus for the rock mass adjacent to the concrete lining
or is the radial stress G0 is the circumferential (hoop) stress se is the hoop strain First, considering the problem as plane strain axisymmetric about the centre of the tunnel, development from the biharmonic equation leads to the standard equations for the radial and hoop stresses in an elastic annulus:
Equations 1 and 2 show that for a r positive, GE is negative. Thus for a compressive positive water pressure inside the tunnel, a concrete lining goes into hoop tension. If there is no external rock pressure to resist lining dilation associated with the hoop tension then the concrete must support this tension on its own. An initial conservative approach is to design the lining on the basis of this condition, that is, no inward compressive pressure external to the tunnel lining.
