Engineering Rock Mechanics

76 lnfad rock: deformabilify, strengfh and failure

be drawn from it. Derive an expression for the uniaxial compressive
strength of rock in terms of the cohesion and angle of internal
friction.

A6.3 The part of the Mohr envelope in which we are interested is shown
below.

Shear

stress

I 1 Normal stress

(r

Compressive strength OC

If we study the geometry of the left-hand half of the Mohr circle and

the failure envelope, we find the following:

(90+ @)I2

c;-4 Jfailure envelope to

(90-@)/2

0,D

The geometry of the lower right-angled triangle then gives us

tan (45 - (4/2)) = c/(ac/2) which, upon expanding and rearranging,

results in a, = 2c/ tan (45 - (4/2)). This relation can be expanded and

rearranged in a number of ways. For example, the friction angle in

terms of the cohesion and uniaxial strength is also given by

tan (4/2) = (ac - 2c)/(ac + 2c).

This shows how we can use typical laboratory test results to confirm

the applicability of the Mohr-Coulomb criterion, or compute unknown

values from known values.

46.4 The linear Mohr-Coulomb envelope with a tensile cut-off sets

a definite limit on the maximal uniaxial tensile strength of a ma-

terial. By considering the largest uniaxial tensile Mohr circle that

can be drawn, determine this tensile strength limit in terms of 0;

and 4.

A6.4 The geometry of the Mohr-Coulomb criterion, together with the

Mohr circles representing the uniaxial compressive and the uniaxial

tensile strengths, are shown below.