Mohr's Circle

Mohr's Circle for 2D

Mohr's circle is a tool for analyzing element stress. The drawing below shows mohr's circle and some of the relavent points of interest.

Given a state of stress, $({\sigma }_x,{\sigma }_y,{\tau }_{xy})$, we can calculate the principle stresses,

${\sigma }_1=\frac{1}{2}({\sigma }_x+{\sigma }_y)+\frac{1}{2}\sqrt{{({\sigma }_x-{\sigma }_y)}^2+4{\tau }_{xy}^2}$

${\sigma }_2=\frac{1}{2}({\sigma }_x+{\sigma }_y)-\frac{1}{2}\sqrt{{({\sigma }_x-{\sigma }_y)}^2+4{\tau }_{xy}^2}$