# Two dimensional stresses / bi-axial stresses

Notations and units:

$\sigma$ = Normal stress, $\tau$ = Shear stress, $\theta$ = Angle

Units of the notations are in SI units.

Normal stress at an angle from the reference plane

$\sigma_{\theta}=\frac{\sigma_{x}+\sigma_{y}}{2}+\frac{\sigma_{x}-\sigma_{y}}{2}\cos2\theta+\tau_{xy}\sin2\theta$

Shear stress at an angle from the reference plane

$\tau_{\theta}=\frac{\sigma_{x}-\sigma_{y}}{2}\sin2\theta-\tau_{xy}\cos2\theta$

Resultant stress

$\sigma_{R}=\sqrt{\sigma_\theta^{2}+\tau_\theta^{2}}$

Angle between normal stress to one plane and resultant stress (angle of obliquity)

$\tan\phi=\frac{\tau_{\theta}}{\sigma_{\theta}}$

Maximum principal stress

$\sigma_{1}=\frac{\sigma_{x}+\sigma_{y}}{2}+\sqrt{\left ( \frac{\sigma_{x}-\sigma_{y}}{2} \right )^{2}+\tau _{xy}^{2}}$

Minimum principal stress

$\sigma_{2}=\frac{\sigma_{x}+\sigma_{y}}{2}-\sqrt{\left ( \frac{\sigma_{x}-\sigma_{y}}{2} \right )^{2}+\tau _{xy}^{2}}$

Angle of maximum principal stress from the reference plane

$\tan2\theta_{p}=\frac{2\tau_{xy}}{\sigma_{x}-\sigma_{y}}$

Angle of minimum principal stress from the reference plane

$\theta_{p} + 90^{\circ}$

Angles of plane where shear stress is maximum

$\theta_{s}, \theta_{s} + 90^{\circ}$

$\theta_{s} = \theta_{p} + 45^{\circ}$

$\tan2\theta_{s}=\frac{\sigma_{y}-\sigma_{x}}{2\tau_{xy}}$

Maximum shear stress

$\tau_{max}=\sqrt{\left ( \frac{\sigma_{x}-\sigma_{y}}{2} \right )^{2}+\tau _{xy}^{2}}$

$\tau_{max}=\frac{\sigma_{1}-\sigma_{2}}{2}$

Maximum normal stress on maximum shear plane

$\sigma'=\frac{\sigma_{1}+\sigma_{2}}{2}$