Page 197 - Introduction to Continuum Mechanics

182 Stress

4.5 Principal Stresses
From Sect. 2B18, we know that for any symmetric stress tensor T, there exists at least three
mutually perpendicular principal directions (the eigenvectors of T). The planes having these
directions as their normals are known as the principal planes. On these planes, the stress vector
is normal to the plane (i.e., no shearing stresses ) and the normal stresses are known as the
principal stresses. Thus, the principal stresses (eigenvalues of T) include the maximum and
the minimum values of normal stresses among all planes passing through a given point.
The principal stresses are to be obtained from the characteristic equation of T, which may
be written:

where

are the three principal scalar invariants of the stress tensor. For the computations of the
principal directions, we refer the reader to Sect. 2B17.

4.6 Maximum Shearing Stress
In this section, we show that the maximum shearing stress is equal to one-half the difference
between the maximum and the minimum principal stresses and acts on the plane that bisects
the right angle between the directions of the maximum and minimum principal stresses.
Let e 1} 62 and 63 be the principal directions of T and let 7j_, T^, T$ be the principal
+n e s tne umt
stresses. If n = niei+n2*2 3 3 * normal to a plane, the components of the stress
vector on the plane is given by

i.e.,

and the normal stress on the same plane is given by