# Von Mises Stress Calculator

Created by Michael Darcy
Reviewed by Steven Wooding
Based on research by
Mises RV Mechanics of solid bodies in the plastically-deformable state Delphenich DH, translator. Nachrichten von der Gesellschaft der Wissenschaften zu Gottingen (November 1913)
Last updated: Nov 22, 2022

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 calculator.

## 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 , 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 2D von Mises stress mathematics, consider visiting the Mohr's circle calculator to ensure you understand the differences 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

Let us begin our quest to learn how to calculate von Mises stress. We'll look at the two-dimensional cases first. As is often the case in sciences, considering only the x- and y- axes simplifies the calculations significantly. Nevertheless, depending on the known variables, you'll need to choose a suitable von Mises stress formula. The options are:

1. General plane stress (or general 2D von Mises stress equation).

Boundary conditions:

• Principal stresses: $\sigma_3 = 0$;
• General stresses: $\sigma_z = 0$; and
• Shear stresses: $\tau_{yz} = \tau_{zx} = 0$.

Notice that since we're considering a 2D von Mises stress calculation first, we just set every quantity related to the z-axis to 0. Hence:

$\scriptsize \sigma_\text{v} = \sqrt{\sigma_x^2 - \sigma_x\sigma_y + \sigma_y^2 + 3\tau_{xy}^2},$

where:

• $\sigma_{\text{v}}$ – Von Mises stress;
• $\sigma_x$ – Normal stress in the x-direction, sometimes denoted $\sigma_{11}$ or $\sigma_{xx}$ ;
• $\sigma_y$ – Normal stress in the y-direction, sometimes denoted as $\sigma_{22}$ or $\sigma_{yy}$; and
• $\tau_{xy}$ – Shear stress XY, sometimes denoted as $\sigma_{12}$.

2. Principal plane stress.

Boundary conditions:

• Principal stresses: $\sigma_3 = 0$; and
• Shear stresses: $\tau_{xy} = \tau_{yz} = \tau_{zx} = 0$ as follows from the definition of principal stress.

It means that we only need to consider the principal stresses $\sigma_1$ and $\sigma_2$ in the von Mises stress calculation.

$\scriptsize \sigma_\text{v} = \sqrt{\sigma_1^2 + \sigma_2^2 - \sigma_1\sigma_2 },$

where:

• $\sigma_1$ - Maximum principal stress; and
• $\sigma_2$ - Intermediate principal stress.

💡 Knowing the normal and shear stresses, you can also use the principal stress calculator to find σ1 and σ2.

3. Pure shear stress.

Boundary conditions:

• Principal stresses: $\sigma_1 = \sigma_2 = \sigma_3 = 0$; and
• Shear stresses: $\tau_{yz} = \tau_{zx} = 0$.

In other words, the only contribution to the 2D von Mises stress is the shear stress XY, $\tau_{xy}$.

$\scriptsize \sigma_\text{v} = \sqrt{3}|\tau_{xy}|$

And if you're thirsty for more knowledge, make sure to visit the shear stress calculator!

## 3D von Mises stress formulae

1. General stress

Boundary conditions:

This is the most general case, so there are no restrictions. Thus, the von Mises stress formula is given by:

$\scriptsize \begin{split} \sigma_\text{v} &= \Bigl(\frac{1}{2}\big[(\sigma_x - \sigma_y)^2 + (\sigma_y - \sigma_x)^2 \\ &\quad+ (\sigma_z - \sigma_x)^2\big] + 3(\tau_{xy}^2 + \tau_{yz}^2 + \tau_{zx}^2)\Bigr)^{\frac{1}{2}} \end{split}$

where:

• $\sigma_3$ – Normal stress in the z-direction, sometimes denoted $\sigma_{33}$ or $\sigma_{zz}$;
• $\tau_{yz}$ – Shear stress YZ, sometimes denoted as $\sigma_{23}$; and
• $\tau_{zx}$ – Shear stress ZX, sometimes denoted as $\sigma_{31}$.

2. Principal stress.

Boundary conditions:

• Shear stresses: $\tau_{xy} = \tau_{yz} = \tau_{zx} = 0$
$\scriptsize \begin{split} \sigma_\text{v} &= \Bigl(\frac{1}{2} \big[(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 \\&\quad+ (\sigma_3 - \sigma_1)^2\big] \Bigr)^{\frac{1}{2}}, \end{split}$

where $\sigma_3$ is the minimum principal stress.

🔎 Although the von Mises stress calculator lets you use the pure shear method in 3D, the formula is precisely the same as in the 2D case.

## How to use the von Mises stress calculator? Von Mises yield criterion

Since the von Mises yield criterion is used to make it easier to use calculations in practice, it's only fitting to sum up by talking about utilizing the von Mises stress calculator. The general instructions are pretty straightforward:

1. Select the appropriate dimensions. While the real world operates in 3D, there are some problems where 2D von Mises stress equations are applicable.

2. Choose the method you want to use. This will depend on the quantities that are known to you and the boundary conditions.

3. Input the stresses as required. For example, for general 2D von Mises stress, you'll be asked to provide $\sigma_x$, $\sigma_y$, and $\tau_{xy}$.

4. You'll see the result of the von Mises stress calculation, $\sigma_\text{v}$.

What can you do with this result? Generally, you will want to compare it to the yield strength (or elastic limit) of a material, $S_\text{y}$. The von Mises yield criterion often takes the form:

$\scriptsize \sigma_\text{v} \geq S_\text{y}$

In other words, if the von Mises stress for a given load is greater than the material's yield strength, it is expected to yield (deform).

## FAQ

### How do I calculate von Mises stress from principal stresses?

To calculate von Mises stress using principal stresses:

1. Determine your principal stress components: σ₁ - maximum, σ₂ - intermediate, and σ₃ - minimal.

• If the problem is in 2D, set σ₃ = 0.
1. Substitute into the von Mises stress equation:

σv = 1/√2 × √((σ1 - σ2)2 + (σ2 - σ3)2 + (σ3 - σ1)2).

• Hint: For a 2D case, it simplifies to:
σv = √(σ12 + σ22 - σ1σ2).
1. You found the von Mises stress, σv!

### When do I use von Mises stress?

Von Mises stress is typically used to predict the behavior of a ductile and isotropic material under complex loading conditions. More specifically, you'd use it to estimate whether the material will yield or fracture by comparing the equivalent stress to the yield value from the tensile test.

### Can the von Mises stress be greater than principal stress?

In short, yes, von Mises stress can be greater than principal stresses. However, it depends on the circumstances. For example, if one of the stresses is compressive, you may obtain such a result.

### What is the von Mises stress for a circular shaft under a torque?

Taking the shear stress to be τxy = 6.5 MPa, the von Mises stress for a circular shaft subjected to torsion is σv = 11.26 MPa. Because there are no other loads, we can use the pure shear con Mises stress formula.

Michael Darcy
Problem conditions
Dimensions
2D (plane)
Method
Principal stresses
Boundary conditions:
Principal stresses: σ3 = 0
Shear stresses: τxy = τyz = τzx = 0
Principal stresses
Maximum principal stress (σ₁)
lbs/ft²
Intermediate principal stress (σ₂)
lbs/ft²
Von Mises stress
Von Mises stress (σᵥ)
lbs/ft²
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