Engineering Rock Mechanics

290 Foundation and slope instability mechanisms

In each case, derive an expression for the factor of safety and, for

the special case of f = 1.0, give an expression for T, p or q as

appropriate.

A 7 7.2 Case (a) Horizontal force
The forces T and W act through the centroid of the block, and the
reaction forces S and N act at the midpoint of AC. However, to simplify
the analysis we assume that they are all coincident with the centroid of
the block, and hence we can ignore moment equilibrium.
For the geometry given, the volume of the block is

and the weight of the block is

Defining the factor of safety for this block as

1 forces resisting sliding

= F,

forces causing sliding

and assuming that the strength of the plane AC is given by the Mohr-

Coulomb criterion, by taking components of the various forces parallel

and normal to AC we obtain

H

F= CLAC + N tan4 - - sin +f

C- + (W cos +f + T sin +f) tan4

W sin +f W sin +f - T cos ef

There are many ways this relation can be rearranged, and one of them

is

2cH + (y H2 cos2 +f + 2T sin2 +f) tan4

(y H2 - 2T) cos +f sin +f

F= (17.1)

For the specific case of c = 0 and T = 0, this reduces to the relation for

Finally, if we examine the case when F = 1, we can rearrange Eq. (17.1)

a block resting on a slope, i.e. F = tan 4/ tan qf.

to obtain

yH2(sin2+ - (1 +cos2q)tanr$) -4cH

2(sin2++(1 -cos2+)tan4)

T= (17.2)