Engineering Rock Mechanics

Questions and answers: in situ rock stress 5 1

and z = down, and take direction n as normal to the desired plane, then

nx =COSanCOS/?nl ny = sina,,cosp,,, n, =sin/?,,.

Note that when the z axis is directed downward, a clockwise bearing
(i.e. measured from North) will always be a positive angle (i.e. anticlock-
wise measured from the x axis) when looking in the direction of the
positive z axis. Hence, the rotation matrix is

1

cos al cos sin al cos sin

COS am COS pm sin am COS /?m sin pm *

Ln, ny nz] ~cosa,cos/?,, sina,,cosp,, sin/?,,]

Now, if we take direction 1 to be a horizontal line on the plane, then
the following relations between the trends and plunges of the axes hold:

=90+a, /?/ =O

am = 180 + a,, pm = 90 - pn

However, when we use these relations, we must ensure that 011 and
am lie between 0" and 360", by subtracting 360" if necessary. Using these
relations, the matrix of direction cosines then becomes

O1

• sin a,, cos a,,


• COS an sin pn COS pn
  cos a,, cos /?,, sin an cos /?,, sin /?,,


• sin a,, sin /?,,

Q4.8 Suppose that we have measured the pre-existing stress state
in the ground by some means and that the results are as follows:
el, magnitude 15 MPa, plunges 35O towards 085O;
u2, magnitude 10 MPa, plunges 43O towards 21 7";
63, magnitude 8 MPa, plunges 27O towards 335O.
Find the 3-D stress tensor in the right-handed xyz co-ordinate
system with
x, horizontal to the east,
y, horizontal to the north,
I, vertically upwards.

the index finger the y-axis and the middle finger the z-axis. A right-handed system is
represented by the digits of the right hand, and a left-handed system is represented by
the digits of the left hand. The use of a left-handed system of axes is convenient for
rock mechanics: the x-axis represents East, the y-axis is North and the z-axis is down.
However, right-handed systems are always used for mathematical work, and in order to
maintain coherence with other mathematical work (as well as the mathematical functions
provided in computers and calculators) we recommend the use of right-handed systems
in rock mechanics. There are two obvious choices for a right-handed system of axes:
x East, y North and z up; or n North, y East and z down. There are advantages and
disadvantages to both of these systems, and so being adept with both is important.
Accordingly, we will use both in this book.