Principal Stress Calculator

What Are Principal Stresses?

Principal stresses are the maximum and minimum normal stresses acting on planes where shear stress is zero. Engineers use them to assess whether a component will fail under combined normal and shear loading.

Principal Stress Formulas

$$\sigma_{\max} = \frac{\sigma_x + \sigma_y}{2} + \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}$$ $$\sigma_{\min} = \frac{\sigma_x + \sigma_y}{2} - \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}$$

The principal plane orientation is $\theta = \frac{1}{2}\arctan\!\left(2\tau_{xy} / (\sigma_x - \sigma_y)\right)$. Maximum shear stress equals the radius term and occurs at 45° from the principal planes.

Design Use

Brittle materials tend to fail along principal planes. For ductile materials, compare combined stress to yield criteria such as von Mises. See also the Mohr Circle Calculator.

Frequently Asked Questions

What is sigma1 versus sigma2?

Sigma1 is the maximum principal stress and sigma2 is the minimum. On the principal planes, shear stress is zero.

Are tau_xy and tau_yx different?

No. For equilibrium they are equal in magnitude. Enter either value in the shear stress field.

What units should I use?

Use consistent stress units such as MPa or psi. All three inputs must share the same unit system.

When do principal stresses matter?

They matter in shaft design, pressure vessels, welded joints, and any case where normal and shear stresses act together.

How does this relate to Mohr's circle?

Principal stresses are the leftmost and rightmost points on Mohr's circle. The circle radius equals maximum shear stress.

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Principal Stress Calculator

What Are Principal Stresses?

Principal Stress Formulas

Design Use

Frequently Asked Questions

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