Page 200 - Introduction to Continuum Mechanics

Maximum Shearing Stress 185

The planes determined by the solutions given by Eq. (ix) are nothing but the principal planes,
on which T s = 0. Thus, on these planes the values of 7J is a minimum (in fact, zero).

The values of if on the planes given by the solutions (x) are easily obtained from Eq. (4.6.4)
to be the following:

and

Thus, the maximum magnitude of the shearing stress is given by the largest of the three values

In other words,

where (r fl) max and (T n) mi n are the largest and the smallest normal stress respectively. It can
also be shown that on the plane of maximum shearing stress, the normal stress is

Example 4.6.1

If the state of stress is such that the components T^, T^ T^ are equal to zero, then it is
called a state of plane stress.

(a) For plane stress, find the principal values and the corresponding principal directions.
(b) Determine the maximum shearing stress.
Solution, (a) For the stress matrix