Free Mohr's Circle Calculator

Mohr’s Circle is an essential graphical tool in solid mechanics for visualizing how stresses transform at a point within a body. This Mohr’s Circle Calculator functions as both a principal stress calculator and a stress transformation calculator, taking the normal stresses ($\sigma_{xx}, \sigma_{yy}$) and the shear stress ($\tau_{xy}$) from a 2D stress state and returning not only the principal stresses but also the maximum shear stress, the orientation angle, the von Mises equivalent stress, and the mean stress. These outputs are critical for evaluating failure criteria, designing mechanical components, and calculating stress concentration factors. The tool is built around the well‑known Mohr’s circle equations, making it quick to use and reliable for both educational and engineering applications.

Understanding the 2D Stress State

At any point inside a loaded body, stresses act in multiple directions. A full three‑dimensional description requires six independent components: three normal stresses and three shear stresses. For many practical problems, however, the out‑of‑plane shear stresses can be ignored ($\tau_{xz} = \tau_{zx} = 0$ and $\tau_{yz} = \tau_{zy} = 0$), reducing the analysis to a 2D stress state characterized by only three components:

• Normal stress acting on the x‑face: $\sigma_{xx}$
• Normal stress acting on the y‑face: $\sigma_{yy}$
• Shear stress on the x‑ and y‑faces: $\tau_{xy}$ (with $\tau_{xy} = \tau_{yx}$ by equilibrium)

These three values completely define the in‑plane stresses. The Mohr Circle Calculator is specifically designed for this planar condition, converting the input into principal stresses and related quantities without requiring manual plotting or algebra.

What Are Principal Stresses?

Principal stresses are the normal stresses that act on planes where the shear stress is zero. They represent the extreme values of normal stress at the point. For a 2D state, the two in‑plane principal stresses – major ($\sigma_{1}$) and minor ($\sigma_{2}$) – are given by:

$$\sigma_{1,2} = \frac{\sigma_{xx} + \sigma_{yy}}{2} \pm \sqrt{\left( \frac{\sigma_{xx} - \sigma_{yy}}{2} \right)^2 + \tau_{xy}^2}$$

From these, the maximum in‑plane shear stress follows as:

$$\tau_{\max} = \sqrt{\left( \frac{\sigma_{xx} - \sigma_{yy}}{2} \right)^2 + \tau_{xy}^2} = \frac{\sigma_{1} - \sigma_{2}}{2}$$

The mean (hydrostatic) stress is the average of the principal stresses:

$$\sigma_{\text{mean}} = \frac{\sigma_{1} + \sigma_{2}}{2}$$

The orientation of the principal planes relative to the original x‑direction is found from:

$$\theta_p = \frac{1}{2} \arctan\left( \frac{2\tau_{xy}}{\sigma_{xx} - \sigma_{yy}} \right)$$

All these results are computed automatically by the principal stress calculator component of the tool. In addition, the von Mises stress is derived from $\sigma_{1}$ and $\sigma_{2}$ using the distortion‑energy theory, providing a yield criterion commonly employed for ductile materials.

Mohr’s Circle: A Graphical Approach to Stress Transformation

Mohr’s circle is a plot of normal stress ($\sigma$) on the horizontal axis versus shear stress ($\tau$) on the vertical axis. It offers an immediate visual representation of how stresses change as the plane of interest rotates. To construct the circle for a given 2D stress state:

1. Place point $A(\sigma_{yy}, \tau_{xy})$ and point $B(\sigma_{xx}, -\tau_{xy})$ on the $\sigma\!-\!\tau$ plane.
2. Connect A and B; this segment is a diameter of the circle.
3. The center lies on the $\sigma$-axis at $\sigma_{\text{avg}} = (\sigma_{xx} + \sigma_{yy})/2$.
4. The radius equals $\tau_{\max}$.
5. Draw the circle. The intersection points with the $\sigma$-axis are exactly $\sigma_{1}$ and $\sigma_{2}$.

The Mohr circle construction is geometrically equivalent to the algebraic principal stress formulas. The calculator performs these steps numerically, giving both the numeric outputs and the conceptual understanding of stress transformation.

Using the Mohr’s Circle Calculator

Operating this 2D stress state calculator is straightforward:

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, $\tau_{xy}$.

The tool immediately returns:

• Major principal stress $\sigma_{1}$
• Minor principal stress $\sigma_{2}$
• Maximum in‑plane shear stress $\tau_{\max}$
• Principal plane angle $\theta_p$
• Mean stress $\sigma_{\text{mean}}$
• Von Mises equivalent stress

All outputs are given in consistent units (e.g., MPa, psi, ksi). The calculator also supports verifying stress transformations for any rotated plane, making it both a maximum shear stress calculator and a stress transformation calculator in one. Whether you are learning stress analysis, verifying hand calculations, or checking finite‑element results, this integrated principal stress calculator and Mohr’s circle generator provides accurate, instant answers.