We have the general expression for shear stress as
Differentiating w.r.t. θ, and equating the derivative to zero,
Since the planes on which maximum shear stresses occur are specific set of planes we may denote them distinctly by Ψ (instead of general aspect angle θ).
Eq. indicates that the planes of maximum shear stress bisect the right angles between the major and minor principal planes.
The normals to the major and minor principal planes may now be defined as the major and minor principal axes. Once the principal stresses and principal planes are known, further analysis may be simplified by expressing the state of stress w.r.t. a new coordinate system with major and minor principal axes as coordinate axes themselves. These axes are usually called axes 1 and 2 respectively.
The general expressions for stress components on arbitrary planes whose aspect angle θ may now be measured with axis-1 as reference axis.
Eq. already defines thatθshould be ± 45º for τnt to be maximum.