Engineering Rock Mechanics

50 In situ rock stress

The values are consistent with each other and they are consistent with

worldwide trends. The value of k = 1 is typical for these depths (noting

that the averaging of the two horizontal stresses masks the fact that the

maximum horizontal stress is a good deal higher than the vertical stress

in both cases), and the value of k reducing with depth commonly occurs.

44.7 How are three-dimensional tensors transformed so that the

stress components on any plane can be calculated?

A4.7 The matrix equation to conduct stress transformation is as follows:

where the stress components are assumed known in the xyz co-ordinate

system and are required in another co-ordinate system Lmn inclined

with respect to the first. The term 1, is the direction cosine of the angle

between the x axis and the I axis. Physically, it is the projection of a

unit vector parallel to Z on to the x axis, with the other terms similarly

defined.

Expanding this matrix equation in order to obtain an expression for

the normal component of stress in the 1 direction, for example, gives

a1 = 1:oxx + l;ayy + 1,2azz + 2(ZxZyaxy + lylzoyz + lzlxad

and a similar expansion for the shear component of stress on the 1 face in

the m direction gives

arm = Ixmx~xx i- Eym,oyy + bwzz + (Exmy + Qnx) axy +

(4% + 4%) ayz + (Lzmx + Lm,) a,,

The other four necessary equations (i.e. for cy, a,, tyz, and tZx) are

found using cyclic permutation of the subscripts in these equations.

So, if we know the orientations of each of the axes in the xyz and

Zmn eo-ordinate systems, together with the stress tensor in the xyz

co-ordinate system, we can calculate the transformed stress tensor.

On many occasions it will be convenient to refer to the orientation of

a plane on which the components of stress are required to be known

using the dip direction/dip angle (a, p) notation. The dip direction is

given as a clockwise bearing from North and the dip angle as an angle

between 0" and 90" measured downwards from the horizontal plane. If

we use a right-handed co-ordinate system6 with x = North, y = East,

' n ' 1 Cyclic permutation is shown graphically to the left.
~==7~

There are many systems of co-ordinate axes used in rock mechanics and it is import-

ant to become fluent in the use of all of them, as with the stress symbols and units.

Three-dimensional co-ordinate axes may be either right-handed or left-handed these are

easily visualized using the thumb, index finger and middle finger of one's hand. If these

three digits are held at right angles to each other, then the thumb represents the x-axis,