Mohr Circle Calculator

Mohr's Circle and Principal Stresses

Mohr's circle is a graphical method to transform 2D stress states and find principal stresses. Given normal stresses σₓ, σᵧ and shear stress τₓᵧ, you can compute maximum and minimum principal stresses and the maximum shear stress.

Principal Stress Formulas

$$\sigma_1 = \frac{\sigma_{xx} + \sigma_{yy}}{2} + \sqrt{\left(\frac{\sigma_{xx} - \sigma_{yy}}{2}\right)^2 + \tau_{xy}^2}$$ $$\sigma_2 = \frac{\sigma_{xx} + \sigma_{yy}}{2} - \sqrt{\left(\frac{\sigma_{xx} - \sigma_{yy}}{2}\right)^2 + \tau_{xy}^2}$$ $$\tau_{\max} = \sqrt{\left(\frac{\sigma_{xx} - \sigma_{yy}}{2}\right)^2 + \tau_{xy}^2}$$

Principal plane orientation: $2\theta = \tan^{-1}\!\left(2\tau_{xy} / (\sigma_{xx} - \sigma_{yy})\right)$.

Frequently Asked Questions

What are principal stresses?

Principal stresses are normal stresses on planes where shear stress is zero. σ₁ is maximum and σ₂ is minimum.

What is maximum shear stress?

τₘₐₓ equals the Mohr circle radius and occurs on planes oriented 45° from the principal planes.

What is von Mises stress?

von Mises stress combines normal and shear components into a single scalar used for ductile yield comparison.

When is Mohr's circle used?

It is used in mechanical design, failure analysis, and geotechnical engineering to evaluate stress states in 2D.

Do τₓᵧ and τᵧₓ differ?

No. For equilibrium τₓᵧ = τᵧₓ. Enter either value in the shear stress field.

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Mohr's Circle and Principal Stresses

Principal Stress Formulas

Frequently Asked Questions

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