Engineering Rock Mechanics

Stress analysis 409

positive shear stresses plot below the o-axis.

2. The trigonometric relations used to simplify the equations resulted in
   = 28:

whatever rotation takes place in real life, twice

the rotation takes place on Mohr's circle.

3. Each point on the circumference of the circle represents the (47) stress
   state on a plane of specific orientation. The points where the circle inter-
   sects the o-axis represent planes on which z = 0: the principal planes.
   The associated o-values are the principal stresses. Mohr's circle shows

the principal stresses are the maximum and

minimum values of normal stress in the body.

4. The points representing the principal planes lie at opposite ends of a
   diameter: in real life planes are perpendicular.


5. The maximum shear stress is given by Y2 (0, - 02) and occurs when
   I$ = 90" (i.e. 8 = 45"). Thus

the planes of maximum shear stress are

orientated at 45" to the principal planes.

Using Mohr's circle to determine principd stresses

1. Draw x-y-axes on the element, draw an element with positive normal
   and shear stresses on it, and so write down (ox, T~) and (o,, zyx).


2. Draw *z-axes (same scale on each) with the o-axis parallel to, and in
   the same direction as, ox. Plot (a,, zq) bearing in mind the positive shear
   stresses plot below the o-axis. Then plot (0,. zyx) on the other side of the
   o-axis. Draw the diameter between the two points, and then draw the
   circle.


3. Calculate the radius as

and the a-value of the centre as l/z(ox + 0,).

4. Calculate the principal stresses and the maximum shear stress:

ol = c + r, o2 = c - r, z,,, = r.