Finite Element Method (FEM) - Rock Mechanics Based on an Anisotropic Jointed Rock Model - page 269

10.5.6 Examples
The excavation of a tunnel is often subdivided into two or more partial excavations
(Section 20.2.1). So the critical construction stages that are decisive for the tunnel's
stability normally arise at stages of excavation before achieving the final state in which
the complete cross-section has been excavated and supported. For a vault heading, the
excavation of the vault can precede that of the bench and invert by 100 m or more and
often is the critical stage of construction.
In the following examples, the results of two-dimensional FEM stability analyses of
tunnels driven according to an advancing vault excavation will be presented (Wittke
1990 and Wittke 2000b).

Advancing vault excavation with open invert
The tunnel considered has an overburden of 40 m and is located in a horizontally bed-
ded (B) and vertically jointed (J) sedimentary rock (Fig. 10.44). The joints J strike par-
allel to the tunnel axis. The vault is supported by a shotcrete membrane with a thickness
of 25 cm. The vault's invert is horizontal and not supported. The mechanical parame-
ters assumed for the rock mass and the shotcrete are listed in Fig. 10.44.
In cases 1 to 3 the influence of the strength of the discontinuities of sets B and J is inves-
tigated. In case 1 the rock mass is assumed to be elastic. In cases 2 and 3, shear param-
eters of c B = 0,

φ J = 20°, respectively, as well as zero
tensile strength normal to the discontinuities are assumed and an iterative viscoplastic
analysis is carried out.
Figure 10.45 shows the normal principal stresses and plastic zones computed for
cases 1 to 3. As a result, the areas above and underneath the vault in which strength
is exceeded in case 1 are far less than in case 3, after carrying out an iterative vis-
coplastic analysis. In case 2 the friction angle that is increased from

φ B = 15°, c J = 0, and φ J = 35° and

φ J = 20° to 35°

leads to considerably smaller plastic zones compared with case 3.
Figure 10.46 (left) shows the vertical displacements along two vertical sections,
one above the vault's roof, the other one above the vault's foot, and along the
vault's arch calculated for cases 1 to 3. In all three cases the vertical displacements
above the vault do not considerably decrease with increasing distance from the
vault. This indicates that the rock mass practically sinks as a block onto the vault.
The vault's feet move outwards and punch into the rock mass underneath the
invert. The displacements of the vault's arch in cases 1 to 3 differ considerably
indicating that the loading of the shotcrete membrane increases significantly with
decreasing shear strength on the discontinuities.

In cases 4 and 5 the effect of reduced lateral stress due to open vertical joints on the
vault's stability is investigated. Open joints are frequently observed in the Bunter
formation in Germany. The reduced horizontal stress is simulated by inducing a
horizontal tensile stress of 100 kPa with horizontal displacements with the aid of
spring elements (cf. Fig. 10.17). This tensile stress is then superimposed on the hori-
zontal compressive stress due to gravity (Fig. 10.46, right). In case 4 the rock mass