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}

COPYRIGHT ©2021 MECHANICAL.IN. ALL RIGHTS RESERVED.

Privacy & Cookie Policy | Terms of Use | User Content PolicyFAQ | Contact us

error: Content is protected !!

Log in with your credentials

Forgot your details?