Mohr's Circle

Given a state of stress, ${\displaystyle (\sigma _{x},\sigma _{y},\tau _{xy})}$, we can calculate the principle stresses,
${\displaystyle \sigma _{1}={\frac {1}{2}}\left(\sigma _{x}+\sigma _{y}\right)+{\frac {1}{2}}{\sqrt {\left(\sigma _{x}-\sigma _{y}\right)^{2}+4\tau _{xy}^{2}}}}$
${\displaystyle \sigma _{2}={\frac {1}{2}}\left(\sigma _{x}+\sigma _{y}\right)-{\frac {1}{2}}{\sqrt {\left(\sigma _{x}-\sigma _{y}\right)^{2}+4\tau _{xy}^{2}}}}$