Page 211 - Numerical Analysis and Modelling in Geomechanics

192 ANNAMARIA CIVIDINI AND GIANCARLO GIODA
The “equivalent” radius of the circular opening is 7 m and the hydrostatic in
situ stress is equal to the unit weight of the rock multiplied by the depth of cover
of the various sections.
The first calibration problem was dealt with by adopting the analytical solution
for a circular hole in an elasto-plastic medium obeying the Mohr-Coulomb yield
criterion, see e.g. [26], This solution leads to the so-called “characteristic curve”
of the tunnel or, in other words, to the relationship between the convergence δ
and the uniform rock pressure p (cf. Figure 6.16).
The possible increment of the rock pressure on the tunnel support due to
possible “loosening” effects was introduced in an approximated manner. In fact,
the pressure obtained with the elasto-plastic solution was increased by the ratio
between a chosen percentage (50%) of the weight of the rock where plastic strain
occurs and the circumference of the tunnel.
Under the assumption of elastic ideally plastic behaviour for the primary
support, three additional parameters are needed to evaluate the stress and strain
regimes developing after its installation, namely:

• its radial stiffness K,
• the limit radial pressure p bearable by the support,
1
• the convergence δ that takes place before its installation.
0
An overall radial stiffness K of 30 MN/m 3 was estimated for the primary
support. This represents the slope of the p–δ curve of the support in Figure 6.16.
The limit pressure p l was determined by assuming, according to the available
2
experimental information, a compression strength of 12 MN/m for the shotcrete
2
and an equivalent tensile strength of 0.22 MN/m for the rock bolts.
The maximum displacement δ of the tunnel wall (representing the input data
m
of the back analysis) depends on the initial displacement δ and on the measured
0
displacement δ* (cf. Figure 6.16),
(6.30)

Since no experimental information was available about the initial displacement
δ , different values of α were introduced in the calculations, namely 1.0 and 1.5,
0
to evaluate its influence on the results of back analyses.
The mentioned elasto-plastic solution permits estimating the long-term radial
displacement of the primary support, and the corresponding average rock
pressure, through the intersection of the characteristic curves of the opening and
of the support.
The second back analysis concerns the determination of the deviatoric
viscosity coefficient η. The numerical model adopted in this case is based on the
finite element approach developed for the analysis of the “squeezing” effects
around tunnels [27].