Von Mises Stress Calculator — Equivalent Stress for 2D & 3D Stress States, Yield Criterion & Factor of Safety | Metric & Imperial

Von Mises Stress Calculator

You may find our von Mises stress calculator useful if you work in engineering or materials science. Did you obtain multiple stresses from your calculations that are hard to use in practice? If so, you're in the right place!

You can use the von Mises stress equation to find the equivalent stress — a single value that you can compare with, for example, the results of tensile tests. Keep reading if you'd like to learn more about the von Mises yield criterion! If you're interested in other ways of telling whether a structure will withstand applied loads, check out the factor of safety concept, which this calculator also estimates for you.

What is von Mises stress?

Von Mises stress is a quantity used to estimate the yield criteria of (usually) a ductile material. It allows you to combine the normal and shear stress components into one equivalent stress.

Think about some goals when designing an object, such as a bridge. We want it to withstand a specific load without breaking the bank to buy the most durable materials known to humanity just in case. But how can we know what will be good enough? Fortunately, we have engineering wisdom and simulations to predict the stresses exerted on our structure. The issue is we often calculate many different stresses. On the other hand, materials most commonly undergo tensile tests, resulting in a single value — a yield strength.

We can't compare the obtained stresses with the material's strength directly. So, what can we do? It's high time the von Mises yield criterion came to the rescue!

🙋 A more rigorous definition of von Mises stress states that it is "the uniaxial tensile stress that would create the same distortion energy as is created by the actual combination of applied stresses."

Before we proceed to the von Mises stress mathematics, it helps to understand the difference between the normal, shear, and principal stresses. Briefly speaking, normal stress is orthogonal to the surface, whereas shear stress acts along a plane. The principal stress is obtained by transforming the current stress state so that the shear stresses vanish.

2D von Mises stress equations

For a 2D (plane) stress state, defined by the two normal stresses σ_x and σ_y together with the shear stress τ_xy, the von Mises stress σ_v is:

σ_v = √(σ_x² − σ_xσ_y + σ_y² + 3τ_xy²)

Equivalently, if you already know the in-plane principal stresses σ₁ and σ₂ (with σ₃ = 0), you can write:

σ_v = √(σ₁² − σ₁σ₂ + σ₂²)

3D von Mises stress formulae

For a general 3D stress state with three normal stresses (σ_x, σ_y, σ_z) and three shear stresses (τ_xy, τ_yz, τ_zx), the von Mises stress is:

σ_v = √{ ½[(σ_x−σ_y)² + (σ_y−σ_z)² + (σ_z−σ_x)²] + 3(τ_xy² + τ_yz² + τ_zx²) }

In terms of the three principal stresses σ₁, σ₂, and σ₃, the same quantity is:

σ_v = √{ ½[(σ₁−σ₂)² + (σ₂−σ₃)² + (σ₃−σ₁)²] }

How to use the von Mises stress calculator?

1. Pick the stress state — 2D plane stress or full 3D.
2. Choose your stress unit — metric (Pa, kPa, MPa, GPa) or American/imperial (psi, ksi). All inputs and results use the same unit.
3. Enter the normal stresses σ_x and σ_y (and σ_z in 3D). Use negative values for compression.
4. Enter the shear stress τ_xy (and τ_yz, τ_zx in 3D).
5. Optionally, enter the material's yield strength to get the factor of safety.
6. Read off the von Mises (equivalent) stress and, if provided, the factor of safety.

Von Mises yield criterion

The von Mises yield criterion states that a ductile material begins to yield when the von Mises stress σ_v reaches the material's yield strength σ_y measured in a simple tensile test:

σ_v ≥ σ_y   →   yielding

The factor of safety against yielding is then FoS = σ_y / σ_v. A factor of safety greater than 1 means the part is (in theory) safe from yielding, while a value below 1 indicates that yielding is predicted.

FAQs

What is the von Mises stress for σx = 80, σy = 20, τxy = 30 MPa?

Using the 2D formula: σ_v = √(80² − 80·20 + 20² + 3·30²) = √(6400 − 1600 + 400 + 2700) = √7900 ≈ 88.88 MPa.

Is von Mises stress always positive?

Yes. The von Mises stress is defined through a square root of a sum of squared differences, so it is always zero or positive, regardless of whether the individual stresses are tensile or compressive.

Can I enter stresses in psi or ksi?

Yes. The von Mises equation is unit-agnostic as long as all inputs share the same unit. Pick metric (Pa, kPa, MPa, GPa) or American/imperial (psi, ksi) from the unit selector, and the result is returned in that same unit.

How does von Mises stress relate to a tensile test?

The von Mises stress is the single uniaxial tensile stress that would create the same distortion energy as your real, multi-axial stress state. That is exactly what lets you compare it directly with the yield strength obtained from a standard tensile test.