Mohr’s circle calculator lets you calculate the principal stresses from a 2D stress state (see stress calculator for more). Utilizing the values of normal stresses and shear stresses on a body, the calculator will return to you the principal stress of the system. The stresses are regarded as one of the most fundamental aspects in designing any body or system (metal, for example). To this end, the calculator incorporates Mohr’s circle equations.

Read on to understand what principal stress is (refer principal Stress Calculator) and how to draw Mohr's circle. The article below contains an example of Mohr's circle. Using this calculator you can calculate — principal stresses — minimum and maximum, maximum shear stresses, angle of orientation, along with von Mises and mean stress. The maximum stress comes in handy to find stress concentration factor. You can also visit our stress concentration factor calculator for more on the topic.

What is a stress state and principal stress?

A stress state of a body is the combination of stresses at a point considering all three directions, i.e., X, Y, and Z or 1, 2, and 3. There are three normal stresses (acting perpendicular to the face) namely, σ11\sigma_{11}, σ22\sigma_{22}, and σ33\sigma_{33} and six shear stresses (acting along the plane) τ12\tau_{12}, τ23\tau_{23}, τ13\tau_{13}, τ21\tau_{21}, τ32\tau_{32}, and τ31\tau_{31},. The stresses acting on the body are shown in the figure below.

Components stress tensor in cartesian system
Components stress tensor in Cartesian system (wikimedia.org)

Considering equilibrium acting on the body, the shear stresses can be reduced to three values, i.e., τ12=τ21\tau_{12} = \tau_{21}, τ13=τ31\tau_{13} = \tau_{31}, and τ23=τ32\tau_{23} = \tau_{32}. Therefore, a stress state can be defined by six stresses, i.e., three normal stresses and three shear stresses. Now, if one considers only in-plane directions, the resultant stress state can be obtained by reducing the stresses, τ13=τ31=0\tau_{13} = \tau_{31} = 0, and τ23=τ32=0\tau_{23} = \tau_{32} = 0. The 2D stress state can now be defined using 3 stresses, i.e., two normal stresses (σ11\sigma_{11}, σ22\sigma_{22}) and a shear stress (τ12\tau_{12}). This can alternatively be shown as given in the figure below (with 1 and 2 directions as x and y).

2D or plane stress state
2D or plane stress state

What is principal stress and How to calculate principal stress?

Consider a state at which only normal stress act on the plane. The stresses at that state are known as principal stresses. This is obtained by transforming the current stress state, i.e., reducing the shear stresses to zero.

2D or plane stress state
2D or plane stress state

Mathematically, the principal stresses can be written using the principal stress equation:

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. Similarly, the maximum shear stress (τmax\tau_\mathrm{max}) for the state can be given by the equation:

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

Alternatively, maximum shear stress can also be defined using the principal stresses as:

τmax=σ1σ22\tau_\mathrm{max} = \frac{\sigma_1 - \sigma_2}{2}

and the mean stress (σmean\sigma_\mathrm{mean}) is written as:

σmean=σ1+σ22\sigma_\mathrm{mean} = \frac{\sigma_1 + \sigma_2}{2}

The angle of orientation, θ is given by:

2θ=tan12τxyσxxσyy\scriptsize 2\theta = \tan^{-1}{ \frac{ 2\tau_\mathrm{xy} } {\sigma_\mathrm{xx} - \sigma_\mathrm{yy}} }

The above set of equations helps you in drawing Mohr's circle and vice versa. The Mohr's circle equations can also be derived or obtained using the geometrical approach as given in the following section.

What is Mohr's circle — Utilizing the Mohr's circle to estimate principal stress

In order to utilize Mohr's circle to estimate principle stress, first, you need to understand what is Mohr's circle and how to draw a Mohr's circle. A Mohr's circle is a graphical representation of a stress state and is used to perform stress transformations. To draw a Mohr's circle for a given 2D stress state with normal stresses (σxx\sigma_\mathrm{xx} and σyy\sigma_\mathrm{yy}) and shear stresses (τxy \tau_\mathrm{xy} and τyx \tau_\mathrm{yx} ):

  1. Plot the coordinates (σyy\sigma_\mathrm{yy}, τxy\tau_\mathrm{xy}) and (σxx\sigma_\mathrm{xx}, τxy\tau_\mathrm{xy}) as points A and B, respectively with σ\sigma as X axis and τ\tau as Y axis.
  2. Join the points, A and B to obtain diameter AB.
  3. Find the center of the circle, O, i.e., the point at which line AB intersects the X-axis.
  4. Draw the circle with the center point as O.
  5. The points at which the circle intersects X-axis are the principal stresses.

How to use Mohr's circle calculator.

Follow the steps below to use the principal stress formula and Mohr's circle calculator.

  1. Enter the normal stress in X direction, σxx \sigma_\mathrm{xx} .
  2. Insert the normal stress in Y direction, σyy \sigma_\mathrm{yy} .
  3. Fill in the shear stress, τxy \tau_\mathrm{xy} .
  4. Mohr's circle calculator will now use the principal stress equations to calculate maximum and minimum principal stresses, maximum shear stress, angle of orientation, von Mises and mean stress.

FAQ

What is a stress state?

A stress state of a body is the combination of stresses at a point considering all three directions, i.e., X, Y, and Z or 1, 2, and 3.

What is Mohr's circle?

Mohr's circle is the 2D graphical representation of the stress state and can be used for the purpose of stress transformation.

What is principal stress?

Principal stresses are defined as the normal stress acting on a plane when there's no shear stress involved.

How to calculate principal stress?

Principal stresses can be calculated using the principal stress formula:

σ₁ = ((σₓₓ + σᵧᵧ) / 2) + √(((σₓₓ - σᵧᵧ) / 2)² + τₓᵧ²)

σ₂ = ((σₓₓ + σᵧᵧ / 2) - √(((σₓₓ - σᵧᵧ) / 2)² + τₓᵧ²)

where σ₁ and σ₂ are minimum and maximum principal stresses.

Rahul Dhari
Normal stresses
Normal stress in X direction (σ_xx)
psi
Normal stress in Y direction (σ_yy)
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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)
rad
Mean stress (σ_m)
psi
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