# Mohr's Circle Calculator

Created by Luciano Mino
Last updated: Dec 07, 2022

The Mohr's circle calculator can help you find the principal stresses from given normal stresses, and shear stresses on the body.

If you're wondering what Mohr's circle is, don't worry. We've paired this calculator with a short text covering everything you need to know about Mohr's circle, including:

• What principal stresses are;
• How to find principal stresses using Mohr's circle;
• How to use this Mohr's circle calculator (with steps); and
• The max shear stress formula.

## Types of stresses - What are the 'principal stressses'?

There are two types of stresses that we can use to describe a body's stress state:

• Normal stresses ($\sigma$): tensile or compression stresses (see our stress calculator) acting perpendicular to any face of the body. The subscript denotes the face on which the stress acts. Tensile stresses are considered positive, while compression is negative.
• Shear stresses ($\tau$): stresses that are coplanar with any face of the body. We use two subscripts to indicate their direction. The first subscript indicates the face on which the stress acts, while the second subscript denotes the direction on that surface. A positive shear causes a clockwise rotation, and a negative shear causes a counter-clockwise rotation.

Principal stresses are the maximum and minimum normal stresses present in an object when subjected to certain forces. They can be found by using what's known as a stress transformation equation, which takes into account all the different kinds of stresses that an object experiences in any given situation.

In an equilibrium state, we can express the stress using six components: $\sigma_xx$, $\sigma_yy$, $\sigma_zz$, $\tau_{xy}$, $\tau_{yz}$, and $\tau_{xz}$.

## Plane stress

Here, we will consider the state where the $z$-axis stress components are all zero.

$\sigma_{zz} = 0$, $\tau_{yz} = 0$, $\tau_{xz} = 0$.

We can represent this 2D stress state with only three components instead of six:

$\sigma_{xx}$, $\sigma_{yy}$, and $\tau_{xy}$.

This simplification allows us to use a special method to visualize the stresses called Mohr's circle.

✅ Take a look at our shear stress calculator too!

## What is Mohr's circle?

Mohr's circle is a graphical method that allows for the visualization of the relationship between normal and shear stresses.

The principle behind Mohr's circle is that it takes what would normally be complex calculations and turns them into simple rotations around an imaginary circle. Determining the circle's center and radius allows you to calculate the principal stresses quickly.

## How to use Mohr's circle

Follow these steps if you want to know how to find the principal stresses using Mohr's circle:

1. Plot the known stress coordinates $A = (\sigma_{xx},\ \tau_{xy})$ and $B = (\sigma_{yy},\ \tau_{xy})$ using the $Y$-axis as the positive shear stress axis and the $X$-axis as the positive normal stress axis.
2. Join both points to get the diameter $AB$.
3. The point where this line intersects the $X$-axis is the center of the circle.
4. Use this point and diameter to draw the rest of the circle $\text{radius} = AB/2$.
5. The points at which Mohr's circle intersects the $X$-axis are the principal stresses.

## How to use our Mohr's circle calculator (with steps)

1. Enter the normal stress in the $X$ direction $\sigma_{xx}$.
2. Enter the normal stress in the $Y$ direction $\sigma_{yy}$.
3. Enter the shear stress $\tau_{xy}$.
4. That's it! The Mohr's circle calculator will now automatically output the principal stresses and the maximum shear stress, along with other useful parameters.

## Principal stresses and max shear stress equations

Alternatively, you can manually find the principal stresses and max shear stress with the following formulas:

\scriptsize \begin{align*} \sigma_1 &= \frac{ \sigma_{xx} + \sigma_{yy}}{2} + \sqrt { \left ( \frac{ \sigma_{xx} - \sigma_{yy}} {2} \right )^2 + \tau_{xy}^2} \\ \sigma_2 &= \frac{ \sigma_{xx} + \sigma_{yy}}{2} - \sqrt { \left ( \frac{ \sigma_{xx} - \sigma_{yy}} {2} \right )^2 + \tau_{xy}^2} \end{align*}

where $\sigma_1$ and $\sigma_2$ are minimum and maximum principal stresses.

And the max shear stress is:

$\scriptsize \tau_\mathrm{max} = \sqrt { \left ( \frac{\sigma_{xx} - \sigma_{yy}}{2} \right )^2 + \tau_{xy} }$

Luciano Mino
Normal stresses
Normal stress in X direction (σ_xx)
psi
Normal stress in Y direction (σ_yy)
psi
Shear stresses
Shear stress (𝛕_xy)
psi
Shear stress (𝛕_yx)
psi
Principal stresses
Maximum principal stress (σ_1)
psi
Minimum principal stress (σ_2)
psi
Other results
Maximum shear stress (𝛕_max)
psi
von Mises stress (σ_mises)
psi
Angle of orientation (θ)
Mean stress (σ_m)
psi
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