Von Mises Stress Calculator

What the von Mises stress calculator finds

A von Mises stress calculator reduces a complex stress state into one number you can compare with the strength of a metal. Enter the normal stress in the x direction, the normal stress in the y direction, and the shear stress, all in psi. The tool returns the equivalent stress in psi.

Picture a bolt that is being pulled and twisted at once. Pulling alone and twisting alone each give a stress, but the material feels both at the same time. The von Mises number is how hard the metal is being pushed, taking all of it together.

The formula in plain words

Von Mises stress = the square root of (σx² − σx × σy + σy² + 3 × τxy²).

1. Square the x stress.
2. Subtract the x stress times the y stress.
3. Add the square of the y stress.
4. Add three times the square of the shear stress.
5. Take the square root.

This is the plane-stress form for a point in a thin body, such as a plate.

Worked example: combined load

Take σx = 15,000 psi, σy = 5,000 psi, and τxy = 3,000 psi.

• σx² = 225,000,000
• σx × σy = 75,000,000
• σy² = 25,000,000
• 3 × τxy² = 27,000,000
• Sum: 225,000,000 − 75,000,000 + 25,000,000 + 27,000,000 = 202,000,000
• Square root: about 14,213 psi

Worked example: pure shear

With only a shear of 10,000 psi and no normal stress, the formula gives the square root of 3 × 100,000,000, which is 17,321 psi.

Common stress states

σx	σy	τxy	Von Mises
20,000	0	0	20,000
0	0	10,000	17,321
10,000	10,000	0	10,000
10,000	−10,000	0	17,321
15,000	5,000	3,000	14,213

A single pull of 20,000 psi gives 20,000. Equal pulls in both directions give 10,000, because the two directions hold each other back. An opposed pull and push give a bigger number, which is why bending and twisting can be hard on a part.

Comparing to yield strength

Yield strength is the stress at which a metal starts to deform for good. A common structural steel has a minimum yield of 36,000 psi, and other grades differ. Check the material certificate or the supplier’s data.

If the von Mises stress from the example above is 14,213 psi and the yield strength is 36,000 psi, the ratio is 36,000 ÷ 14,213, about 2.5. That is a factor of safety against yielding. The factor you need depends on the code and how bad a failure would be.

Where the stresses come from

The three inputs are not measured directly in most cases. They come from other calculations, such as bending stress from a moment, shear from a force on a section, or hoop stress in a pressurized pipe. You can use the bending stress calculator and the shear stress calculator to find them, then combine them here.

Always use the stresses at the same point in the part. Bending stress is highest at the outer fiber, and shear is highest at the neutral axis, so they may not peak together.

Common mistakes

• Dropping the signs. Tension is positive and compression is negative, and the cross term depends on it.
• Using stresses from different points. The combination only makes sense at one location.
• Comparing to ultimate strength. The comparison is to yield strength.
• Applying the 2D form to a point with a third stress. This tool is for plane stress.
• Treating the output as a design approval. Fatigue, stress concentrations, and connections all matter, and a licensed engineer should review structural work.

Related calculators

Start with the stress calculator for force over area. The bending stress calculator and shear stress calculator supply the normal and shear inputs for this page.