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Principal stresses and Mohr's circle

At a point in plane stress there are directions where the shear stress is zero. The normal stresses in those directions are the principal stresses, the largest and smallest normal stress at the point. Mohr's circle is a graphical way to find them, with center and radius from $\sigma_x$ , $\sigma_y$ and $\tau_{xy}$ .

\sigma_{1,2} = \dfrac{\sigma_x+\sigma_y}{2} \pm \sqrt{\left(\dfrac{\sigma_x-\sigma_y}{2}\right)^2+\tau_{xy}^2}

principal stresses

Symbols

$\sigma_1,\ \sigma_2$	largest/smallest principal stress	Pa
$\tau_{xy}$	shear stress	Pa

Example

$\sigma_x = 80$ MPa, $\sigma_y = 20$ MPa, $\tau_{xy} = 30$ MPa:

center $= 50$ , radius $= \sqrt{30^2+30^2} \approx 42.4$ , so $\sigma_1 \approx 92.4$ MPa.

The center of Mohr's circle is always the average

(\sigma_x+\sigma_y)/2

, regardless of the shear stress.

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