Suppose that normal and shear stresses are known with respect to two orthogonal coordinate axes $\mathrm{x}$ and $\mathrm{y}$. Strains with respect to different orthogonal axes $\mathrm{n}$ and $\mathrm{t}$ oriented at an angle $\mathrm{\theta }$ relative to the $\mathrm{x}$ axis may be determined from the equations:

${\sigma }_n=\frac{{\sigma }_x+{\sigma }_y }{2}+\frac{{\sigma }_x-{\sigma }_y }{2}\cos 2\theta + {\tau }_{xy}\sin 2\theta$

${\sigma }_t=\frac{{\sigma }_x+{\sigma }_y }{2}-\frac{{\sigma }_x-{\sigma }_y }{2}\cos 2\theta - {\tau }_{xy}\sin 2\theta$

${\tau }_{nt}=-\frac{{\sigma }_x-{\sigma }_y }{2}\sin 2\theta + {\tau }_{xy}\cos 2\theta$

There exists one set of axes for which the normal stresses are maximum and minimum. These axes are called the principal axes. The values of the maximum and minimum normal stresses can be computed from:

${\mathrm{\sigma }}_{p1},{\mathrm{\sigma }}_{p2}=\frac{{\mathrm{\sigma }}_x+{\mathrm{\sigma }}_y}{2}\pm \sqrt{{\left(\frac{{\mathrm{\sigma }}_x-{\mathrm{\sigma }}_y}{2}\right)}^2+{{\tau }_{xy}}^2}$

Relative to the $\mathrm{x}$ and $\mathrm{y}$ axes, the principal axes are oriented at an angle given by:

$\tan 2{\theta }_p=\frac{2{\tau }_{xy}}{{\sigma }_x-{\sigma }_y}$