Expressions of Principal Planes and Principal Stresses:

In calculus, you have learnt that while a function reaches maximum or minimum its derivative along with respect to the independent variable becomes zero. Because the normal stress on an arbitrary plane is a function of the aspect angle θ as given by the expression,  σn = σx+σy/2 +σx-σy/2  cos 2θ + τxy sin 2θ, the maxima and minima of σn occur on the planes for which dσ_n/dθ becomes zero, (similarly, τ_nt will be maximum on planes where dσ_nt  = 0).

Let us now derive the expression,

dσ_n/dθ   = σ_x - σ_y/2 (-2 sin 2θ) + τ_xy2 cos 2θ

=2(τxy cos 2θ +(σy -σx/2)sin 2θ) = 2τ_nt

i.e. dσ_n/dθ   = 2τ_nt                                                                                                                                  . . . (1)

Eq. (1) provides important features of the principal plane, namely, the absence of shear stress components on the plane. We can, thus, alternatively describe a principal plane as a plane on which only a general stress component is acting. While dealing with a three-dimensional state of stress you will search that the third principal plane is neither maximum nor minimum. Hence, we will describe principal planes as planes on that shear stresses are zero.

 Equating dσ_n/dθ    to zero we get

Τ_xy cos2θ +σ_y -σ_x/2 sin 2θ =0

Or sin 2θ/cos2θ = -2τ_xy/σ_y-σ_x = 2τ_xy/σ_x-σ_y

Denoting the specific angles defining principal planes by σ1 and σ2,

 or       tan 2Φ = 2τ_xy/σ_x-σ_y                                                                                                                  . . . (2)

Eq. (2) gives a condition for the determination of principal planes. Eq. (2) will have two solutions within the range -π/2 < Φ < π/2 and they will give the orientation of principal planes.