Stability of Rock Wedges and Excavation Surfaces - Rock Mechanics Based on an Anisotropic Jointed Rock Model - page 304

(11.40)

(11.41)

The state of limit equilibrium on the internal failure lines can then be formulated as

(11.42)

where Di , φ i,d and c i,d are the lengths and the design values of the shear parameters of

the internal failure lines. If the internal failure lines are formed in the intact rock, φ i,d

and c i,d correspond to φ IR and c IR . Seepage forces, which may act on the internal failure

lines, are considered by line forces W Ii . Solving (11.42) for tanϕi ϕ i yields:

(11.43)

Equations (11.35) and (11.43) form a coupled nonlinear system of 2n equations for the
unknowns i and ϕ i , and

ϕ i are determined from (11.43) by means of an iterative procedure. In the special case of
c i,d = 0 and W Ii = 0, the angles

ϕ i (i = 1,...,n). i are determined from (11.35) as a function of

ϕ i are equal to the friction angles φ i,d and the unknowns

i can be directly solved from (11.35).
To fi nd out the most unfavorable confi guration of external and internal failure lines of
a given rock mass leading to the maximum required anchor force we have to vary the
orientations of the failure lines.

Simplifi ed methods for the stability analysis of two-dimensional sliding masses on poly-
gonal surfaces originally developed for soil can also be applied to rock. Such methods are
the slice method according to Janbu (1954) and the general wedge method. These methods
are formulated according to the partial safety factor method in DIN 4084-2009-1 (2009)
and have been implemented, for example, in the software GGU-STABILITY (GGU 2011).
Janbu's slice method requires the subdivision of a potential sliding mass into vertical slice-
shaped sections bounded by fi ctitious internal section lines. Along the latter only normal
forces are applied and no shear resistance is assumed (Fig. 11.15, above). This assump-
tion leads to conservative results compared with the method described above. The shear
parameters along the polygonal failure line can be separately specifi ed for each slice. For
example, the wedge illustrated in Fig. 11.15 (upper), which is supported by two disconti-
nuities D1 and D2 exhibiting the shear parameters c D1 , φ D1 , c D2 and φ D2 , is subdivided

into slices. The utilization factor μ J is calculated as the sum of the quotient of driving and

resisting shear forces acting parallel to the failure line.
In the general wedge method, a two-dimensional wedge supported by a polygonal failure
line is subdivided into two or more blocks bounded by the failure line and internal failure
lines. The shear parameters of each block can be chosen differently. Along the internal
failure lines the mean shear strength of the adjacent blocks is assumed (Fig. 11.15, low-