Engineering Rock Mechanics

122 Rock masses: deformabili% strength and failure

8.2 Questions and answers: rock masses

48. I For a simple sedimentary rock mass in which the only effective
    fractures are the bedding planes, the elastic modulus of the rock
    mass can be found from the addition of the displacements due to
    both the intact rock and the fractures, noting that the rock mass can
    comprise more than one stratum, each containing bedding plane
    fractures with different frequencies.
    Unfrucfured struta Consider firstly the case of n strata of intact
    rock, each with a thickness ti and modulus of elasticity Ei. Derive
    an expression for the composite elastic modulus, E,, of the rock
    mass in a direction normal to the strata by considering the total
    displacement (and hence strain) of the total thickness of the rock
    mass due to the applied stress. Write the expression in terms of €1
    and ti, and assume that the interfaces between adjacent units have
    no mechanical effect.
    Struta with bedding plane fracfures. Now consider the case where
    each stratum of rock contains a set of bedding plane fractures par-
    allel to the stratum boundaries. The fracture frequency of the set
    within each stratum is unique - stratum i possesses a frequency 1;;
    similarly, the modulus of deformation (i.e. applied stress/displace-
    ment) within each stratum is unique and for unit i is Ed;. Extend the
    expression for E, to include ti, Ei, ki and Ed;.

AS. I Unfractured strata

The model comprises n strata of rock, each of thickness t, and elasti-

city modulus E,. The total thickness of rock is L = Ci tl, where the

summation in this and succeeding equations is from 1 to n.

= &/ti = o/Ei and hence the

displacement of each unit is 6j = tiEi = ti(a/Ei).

1"

The strain in each unit is given by

This expression for Si then gives the total displacement as

?

n 0
ST = CSz = Eti-.
to : 'i 1 Ei

The overall strain is E = ST/L, and the results we have derived here for

SI then give the overall strain as

E=-==.

L Cti

i

Thus, the expression for the rock mass modulus is

ti

i

Note that the ti cannot be cancelled in the numerator and denominator

of the right-hand side of this equation, as is evident when the expression