Von Mises Stress Calculator

Calculate the von Mises equivalent stress from a full 3D or plane stress state. Enter the six stress components below to get the equivalent stress, principal stresses, maximum shear, and (optionally) the yield utilisation and factor of safety.

What is Von Mises Stress?

Von Mises stress (also called equivalent stress or effective stress) collapses a complex multi-axial stress state into a single scalar that can be compared against the uniaxial yield strength of a material. It is based on the distortion energy theory: yielding of a ductile metal begins when the energy of distortion reaches the same value as at yield in a simple tension test.

Because it ignores hydrostatic (volumetric) stress and responds only to the shape-changing part of the stress tensor, the von Mises criterion is the workhorse failure measure for steel, aluminium, and other ductile materials, and it is the default stress contour in most finite element packages.

Von Mises Stress Formula

General 3D Stress State

This is the most general form, taking all three normal stresses and all three shear stresses. It can equivalently be written in terms of the principal stresses σ₁, σ₂, σ₃.

In Terms of Principal Stresses

Plane Stress (2D)

For a thin plate or shell where the out-of-plane stresses are zero, the formula simplifies to the expression above. This calculator handles the plane stress case automatically: just leave σ_z, τ_yz, and τ_zx set to zero.

Yield Check and Factor of Safety

Once the equivalent stress is known, the yield check is a direct comparison with the material yield strength:

If you enter a yield strength, the calculator reports both. A utilisation at or below 100% means the point is elastic; above 100% means the von Mises criterion predicts yielding. Typical yield strengths are around 250 to 355 MPa (36 to 50 ksi) for structural steel and 240 to 280 MPa for common aluminium alloys.

Von Mises vs Tresca

The Tresca (maximum shear stress) criterion predicts yield when the maximum shear stress τ_max = (σ₁ − σ₃)/2 reaches f_y/2. Equivalently, the Tresca equivalent stress σ₁ − σ₃ is compared directly with f_y. Tresca is always more conservative than von Mises, by up to about 15% in pure shear. Von Mises matches experimental yield data for ductile metals more closely, which is why it is the more common choice. This calculator reports both the absolute maximum shear stress and the Tresca equivalent stress alongside the von Mises value so you can compare them.

Notes on Accuracy

The principal stresses are found by solving the characteristic equation of the symmetric stress tensor exactly, so the reported σ₁, σ₂, σ₃ and the resulting von Mises and maximum shear values are exact for the stress state you enter. The only assumption is that the input represents a single material point. To recover the stress state at every point of a real structure, including stress concentrations and load combinations, use a full finite element model.

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