Engineering Rock Mechanics

58 In situ stress

Axial strain, E, = &= 1. -

Young’s modulus,

E=%

Ea

, “a

I

“a

Lateral strain.

Poisson’s ratio.

1 “v

Figure 4.13 Strains on a small element of rock. (a) Axial strain and Young’s

modulus. (b) Lateral strain and Poisson’s ratio. (c) Vertical and horizontal strains.

In this case, the total strain along any given axis may be found from the

strain due to the associated axial stress, with the induced strain

components due to the two perpendicular stresses being subtracted.

For example, the vertical strain, q, is given by the expression

where oH1 and OH2 are the two principal horizontal stress components.

In the same way, the horizontal strain, can be expressed as

To provide an initial estimate of the horizontal stress, we make two

assumptions:

(a) the two horizontal stresses are equal; and

@) there is no horizontal strain, Le. both

We began this analysis by considering an element within an isotropic rock

mass, and so we would expect the two horizontal stresses induced by the

vertical stress to be equal. Moreover, the element of rock cannot expand

horizontally because it is restrained by similar adjacent elements of rock,

each of which is also attempting to expand horizontally. If, therefore, we

take EH~ as zero in the second equation above we find

and EH2 are zero.