Instability of Rock Masses - Environmental Geomechanics - page 96

The principle of the stability calculation is based on a double verification of the
equilibrium conditions of the sliding or toppling blocks, going from upstream to
downstream. As an example, let us consider the group of blocks in Figure 4.5, with
constant thickness ∆ and variable height h i , whose base is inclined at an angle α
less than the angle of frictionφ . The weight of each boulder is W=γ∆xh .

i i

The first boulder with the tendency to topple over ( h n / ∆ x > cosα) exercises on a
summit force P n the lower adjacent boulders parallel to its base and an interslice
force P n tg φ. The condition of equilibrium gives:
(

)

h sin α −∆ x cos α

n

P= W [4.7]

n n

2 h

n

Subjected to these forces, the block of order n + 1 could slip or topple over. The
equilibrium conditions are verified by calculating the force P n+1 transmitted to the
following block:

cos α tgj- α sin

- sliding P= P-W [4.8]

n+1 n n+1

2

1-tg α

( ) ( )( )

Pz −∆ tg φ + W /2 h sin ∆ cos α − x α

nn + 1 n +1 n + 1

- toppling P = [4.9]

n +1

y

n + 1

with z n +1 , y n+1 : heights of the loads P n , P n +1 on the rotation center of the block n + 1.

Figure 4.5 . Model of cliff with several blocks, according to [HOE 81]