Stability of Rock Wedges and Excavation Surfaces - Rock Mechanics Based on an Anisotropic Jointed Rock Model - page 327

hill to each block i are calculated successively assuming limit equilibrium. The contact
force If i-1 is obtained by summing up all moments about the axis of rotation.
When checking the stability against sliding (Fig 11.32, right), the contact forces Ii-1 i-1 and
I i are determined from the balance of forces parallel to the sliding and contact surfac-
es in a corresponding manner as described above for the case of sliding on polygonal
surfaces.

Because it is not known a priori whether a single block fails due to toppling or sliding
the contact forces Ii i have to be determined for both cases. The greater one is applied
to the adjacent block downhill. This procedure has to be progressively repeated as far
as the lowest block. The result of the equilibrium condition for the lowest block is the
force P 0 (Fig. 11.31, left) that is calculated by replacing φ d . If

P 0 ≤ 0, the slope is stable. If P 0 > 0, a supporting anchor force A = P 0 is required to
stabilize the slope.
The method of Goodman & Bray (1976) has been further developed by Zanbak (1983),
Aydan et al. (1989) and Kliche (1999). Also, numerical procedures such as the distinct
element method (DEM) have been applied to the stability analysis of multiple rock
blocks (Cundall 1971, Ishida et al. 1987, Lanaro et al. 1997, Eberhardt et al. 2004). In
cases where the thickness of blocks is small compared with the height of the blocks,
toppling of a slope was proposed to be analyzed by a continuous approach using ana-
lytic solutions (Bobet 1999, Sagaseta et al. 2001, Liu et al. 2008).
The stability against toppling of rock blocks or columns, which are bounded by
non-persisting discontinuities, was investigated by Majdi & Amini (2008) by determin-
ing the distribution of tensile stresses acting in the rock bridges between the columns
caused by the moment due to self-weight using both solid mechanics and fracture me-
chanics concepts.

φ by the design value

11.6 Stability of Rock Columns and Layers against Buckling

The FEM and the mechanics of a rigid body do not allow solving stability problems
which are caused by buckling and bulging. These phenomena can lead to failure of
unsupported rock columns and layers adjoining excavation surfaces of rock structures
(Fairhurst & Cook 1966).

Figure 11.33 shows a cavern in a rock mass separated by two vertical discontinuity sets
D1 and D2 that strike perpendicular (D1) and parallel (D2) to the cavern axis, and a
horizontal discontinuity set D3. The cavern's wall and the discontinuities define the
sides of a rock column of width s 1 , thickness s 2 and height h. The column is loaded
in the vertical direction by the stress

σ z while the horizontal stress σ y is approximately

equal to zero provided that the cavern's wall is not supported.
The rock column by approximation can be regarded as a beam that is simply supported
at its lower and upper ends and axially loaded by the compressive stress

σ z (Fig. 11.33).
The lateral yielding of a beam axially loaded by a compressive stress is referred to as
buckling. According to the elastic beam theory of second order, the buckling force of a
simply supported beam (Euler case 2) is given as (Cavers 1981)