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 (σ\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 (τ\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: σxx\sigma_xx, σyy\sigma_yy, σzz\sigma_zz, τxy\tau_{xy}, τyz\tau_{yz}, and τxz\tau_{xz}.

Plane stress

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

σzz=0\sigma_{zz} = 0, τyz=0\tau_{yz} = 0, τxz=0\tau_{xz} = 0.

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

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

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

  1. Enter the normal stress in the XX direction σxx\sigma_{xx}.
  2. Enter the normal stress in the YY direction σyy\sigma_{yy}.
  3. Enter the shear stress τxy\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:

σ1=σxx+σyy2+(σxxσyy2)2+τxy2σ2=σxx+σyy2(σxxσyy2)2+τxy2\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 σ1\sigma_1 and σ2\sigma_2 are minimum and maximum principal stresses.

And the max shear stress is:

τmax=(σxxσyy2)2+τxy\scriptsize \tau_\mathrm{max} = \sqrt { \left ( \frac{\sigma_{xx} - \sigma_{yy}}{2} \right )^2 + \tau_{xy} }

🔎 See our shear modulus calculator to learn more about shear stress and shear strain.

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