Engineering Rock Mechanics

368 Underground excavation instability mechanisms

Thus, we see that the first of these angles represents the location where
the rock is subjected to the greatest compressive stress, and the second is
the location where it is subjected to the greatest tensile stress.
In order to determine the compressive strength of the material, we use
the relation
2c
a, =
tan (45 - $)
which was derived in A6.3 to find

= 128.7 MPa.

2.30 60

a, =

tan 25

Thus, as the compressive strength of the rock is 128.7 MPa and its tensile

strength is zero, we can see that the excavation boundary is stable at the

location of maximum compressive stress, but unstable at the location of

greatest tensile stress.

It is important to realize that the parameter x is the angle the tangent

to the boundary of the opening makes to the horizontal. A more useful

parameter for locating position is the angle a point subtends to the major

axis, and this is shown by 6 in the sketch above. From the geometry of

an ellipse, we find that these angles are related through the relation

1

tme = -

4 - B)

although, in order to take account of the correct quadrant of the tri-

gonometrical functions, the angle 0 is better evaluated using the atan2

function

0 = atan2 [-q sin()( - B), COSO( - B>].

Using this relation, we find that the locations of the points of max-

imum and minimum boundary stress relative to the major axis of the

ellipse are then

Omax = atan2 [-2.5 sin(-20.3" - 45"), cos(-20.3" - 45")]

= atan2(2.271, 0.418) = 10.4"

and

Omax = atan2 [-2.5 sin(69.7" - 45"), cos(69.7" - 45")]

= atan2(-1.045, 0909) = 139.0"

The following diagram shows the complete boundary stress distribu-

tion, together with the locations of the maxima and minima. The lines

drawn normal to the boundary represent the stress magnitude at that

position, with lines inside the excavation profile indicating tensile stress.

A useful technique for qualitatively assessing stress distributions is

the so-called 'streamline analogy', whereby we picture the opening as an

obstruction standing in flowing water. For the case under consideration

3Programming languages such as Fortran include the atan2 function in order to

resolve the issue of correct quadrant. The same effect can be obtained on a calculator

using 'rectangular to polar' co-ordinate conversions.