Expressions of Principal Planes and Principal Stresses:

In calculus, you have study that when a function attain maximum or minimum its derivative w.r.t. the independent variable becomes zero. As the normal stress on an arbitrary plane is a function of the aspect angle q as given by the expression,

σ _n = ((σ _x + σ _y) / 2) +(( σ _x - σ _y)/2 ) cos 2q + t_xy sin 2q, the maxima and minima of σ_n occur on the planes for which d σ _n / dq becomes zero, (similarly, t_nt will be maximum on plane where d t _nt/ dq = 0).

Let us now derive the expression,

d σ _n  / dq = (σ _x - σ _y)/2 (- 2 sin 2q) + t_xy 2 cos 2q

=  2 (t _xy cos 2q + ( (σ _y - σ _x)/2) sin 2q

= 2 t_nt

That means

d σ _n / dq  = 2 t_nt       

Eq. gives a significant characteristic of the principal plane, such as, the absence of shear stress components on the plane. Therefore, we can alternatively describe a principal plane as a plane on which just a normal stress component is working. While dealing with a three-dimensional state of stress you shall find that the third principal plane is neither maximum nor minimum. Therefore, we shall define principal planes as planes on which shear stresses are zero.