Page 124 - Rock Mechanics For Underground Mining

ROCK STRENGTH AND DEFORMABILITY

Figure 4.23 Coulomb strength en-
velopes in terms of (a) shear and nor-
malstresses,and(b)principalstresses.

Applying the stress transformation equations to the case shown in Figure 4.22
gives

1
1
n = ( 1 + 3 ) + ( 1 − 3 ) cos 2
2 2
and
1
= ( 1 − 3 ) sin 2
2
Substitution for n and s = in equation 4.11 and rearranging gives the limiting
stress condition on any plane defined by as

2c + 3 [sin 2 + tan (1 − cos 2 )]
1 = (4.12)
sin 2 − tan (1 + cos 2 )
There will be a critical plane on which the available shear strength will be first
reached as 1 is increased. The Mohr circle construction of Figure 4.23a gives the
orientation of this critical plane as

= + (4.13)
4 2
This result may also be obtained by putting d(s − )/d = 0.
For the critical plane, sin 2 = cos , cos 2 =−sin , and equation 4.12 reduces
to

2c cos + 3 (1 + sin )
1 = (4.14)
1 − sin
This linear relation between 3 and the peak value of 1 is shown in Figure 4.23b.
Note that the slope of this envelope is related to by the equation
1 + sin
tan = (4.15)
1 − sin
and that the uniaxial compressive strength is related to c and by
2c cos
c = (4.16)
1 − sin
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