Squeezing Rock - Rock Mechanics Based on an Anisotropic Jointed Rock Model - page 97

Figure 4.7 Yielding steel set (Bochumer Eisenhütte Heintzmann GmbH & Co. KG)

4.4 Convergence-Confinement Method

The convergence-confinement method is based on closed analytic solutions for an in-
finite weightless disk with a circular hole in its center, representing the tunnel loaded
with uniform normal stresses p r and p 0 , acting on its inner and outer boundaries, respec-
tively (Fig. 4.8). Plane strain conditions and isotropic elastic-plastic stress-strain behav-
ior of the disk representing the rock mass are normally assumed. Thus, isotropic elastic
deformability and strength are adopted, and the plastic, irreversible displacements are
independent of time (Egger 1973, Ribacchi & Riccioni 1977, Oreste 2003, Oreste 2009).
Thus, a rotationally symmetric solution is obtained for the radial displacement δ r at the

boundary of the hole representing the tunnel's contour and the plastic zone bounded
by the plastic radius R pl (Fig. 4.8). The supporting pressure p r as a function of radial
displacement δ r is referred to as the “ground reaction curve” (Fig. 4.9).

The radial displacement

δ r of the support, uniformly loaded on its outer boundary with
p r , is determined on the basis of the elastic plane strain solution of a hollow cylinder
(Jaeger & Cook 1979). The so-called “support reaction curve” is then obtained in the
same way as the ground reaction curve by plotting the supporting pressure p r as a func-
tion of radial displacement δ r (Fig. 4.9).

The intersection point of the ground reaction curve and the support reaction curve
defines the supporting pressure and the radial displacement at equilibrium.
If a stiff lining is installed immediately after excavation of the tunnel, a very high radial
supporting pressure p r0 is required to stabilize the tunnel (Figs. 4.3 and 4.9). This pres-