Engineering Rock Mechanics

108 Intact rock

BASIC EQUATIONS Rock fails at a critical combination of normal and shear stresses:

IT1 = To + pun

z0 = cohesion I*. = coeff. of friction

171 =+(u, - u3) sin 2p

U" =i(LT, + UJ +i(u, -a3) cos 2p

The equation for 171 and on are the equations of a circle in

(u, T) space:

FUNDAMENTAL GEOMETRY

Ten;m cutoff, To Mohr envelope

ut Dl

Uniaxial

tension compression

At failure,

2p = 90 + I$

' 1 2P j p = 45 + f

Un2xial

Figure 6.18 The Mohr-Coulomb failure criterion.

the linear Mohr envelope, which defines the limiting size for the Mohr's

circles. In other words, (FZ co-ordinates below the envelope represent

stable conditions; CFZ co-ordinates on the envelope represent limiting

equilibrium; and o-zco-ordinates above the envelope represent conditions

unobtainable under static loading. Because the criterion is developed for

compressive stresses, a tensile cut-off is usually utilized to give a realistic

value for the uniaxial tensile strength.

We anticipate that this criterion is most suitable at high confining

pressures when the material does, in fact, fail through development of

shear planes. At lower confining pressures, and in the uniaxial case, we

have seen that failure occurs by gradual increase in the density of

microcracks sub-parallel to the major principal stress, and hence we would

not expect this type of frictional criterion to apply directly. However, at the

higher confining pressures, the criterion can be useful and it should be

noted, with reference to Fig. 6.18, that the failure plane will be orientated

at p = 45" + (qY2).

The influence of a significant water pressure in porous materials (which

is deducted from the normal stress components, but not from the shear

stress component) is clear as the Mohr's circle is moved to the left by an

amount equal to the water pressure, hence introducing the possibility of

the Mohr's circle moving from a stable region to be in contact with the

Mohr envelope.

Despite the difficulties associated with application of the criterion,

it does remain in use as a rapidly calculable method for engineering

practice, and is especially significant and valid for discontinuities and

discontinuous rock masses.