Equation for principal stresses and principal planes:

Derive equation for principal stresses and principal planes for the element subjected to compound stresses.

Sol.: For state of stress shown in the given figure the normal stress and shear stress on any oblique plane inclined at angle θ is determined by,
 
 
As the principal plane should carry normal stress, shear stress acting on it is to be zero.
That is,
    
This provides one of the principal planes, other principal plane shall be perpendicular to this plane.

Substituting value of θ in σ_n expression by rearranging as below.
 
It provides the values of principal stresses. The largest one is called as major and smaller is called as minor principal stress.