Engineering Rock Mechanics

Design against stress-controlled instability 385

Following this procedure allows the production of three ground response

curves (one each for the floor, sidewall and roof) in radial stress-radial

displacement space. In itself, such a ground response curve is of limited use:

to be of utility to an engineer, it is important to see how the ground

response curve interacts with the curve representing the behaviour of a

supporting element (see Brady and Brown, 1985, for more information).

Available support lines. For all elements that are used for reinforcement or
support, it is possible to determine (either using closed form solutions, or
by numerical calculation) the radial stress-radial displacement behaviour
of the support system. To illustrate this, a concrete lining, as an example,
provides a specific support line depending on its geometry and material
properties. These support lines are commonly known as available support
lines.
In Fig. 20.27, there is a cross-section through a shotcrete or plain concrete
circular lining. By approximating this lining to a thick-walled elastic
cylinder subject to an external pressure, a standard solution can be used
to determine the radial stiffness of the lining, kcon, and hence to determine
the available support line through application of the formula pi = kconpi,
where pi is the support pressure and ui is the support displacement. Such
a lining has a maximum strength, and thus the maximum radial stress that
the lining can withstand without crushing is also required. The terms in
the formulae in Fig. 20.27 are: E,,, = Young's modulus of the shotcrete or
concrete; vco,, = Poisson's ratio of the shotcrete or concrete; t,,, = lining
thickness; ri = internal tunnel radius; and ocmm = uniaxial compressive
strength of the shotcrete or concrete.
There are many different types of supporting elements-for example,
blocked steel sets, rock bolts and other types of anchor-and support
stiffness formulae can be established for all these. To present the full range
of the associated formulae is beyond the scope of this book, but interested
readers are referred to Hoek and Brown (1980) for a more comprehensive
list. Using such formulae, the support pressure associated with a given
ground response curve can be explicitly evaluated, and in Figure 20.28, the
available support lines for five different types of support are shown, in
conjunction with the ground response curves for the roof, sidewall and
floor of a tunnel.

Support stiffness:

Maximum support pressure:

2

Figure 20.27 Formulae for the available support line of a shotcrete or plain concrete

circular excavation lining.