Mohr's Circle Calculator

About the Mohr's Circle Calculator

Mohr's circle is a graphical method for analyzing 2D stress states, transforming normal and shear stresses as an element is rotated. Given σx, σy, and τxy, it determines principal stresses (σ₁, σ₂), maximum shear stress, and the orientation of the principal planes.

This calculator computes the center C = (σx + σy)/2 and radius R = √[((σx−σy)/2)² + τxy²], giving principal stresses σ₁ = C + R and σ₂ = C − R. It also calculates stresses on any rotated plane, Von Mises equivalent stress, Tresca maximum shear, and safety factors against yielding.

The tool handles classic load cases—uniaxial tension, pure shear, biaxial equal, pressure vessel, and general plane stress—through preset buttons. A rotation angle input lets you find stresses at any orientation. Engineers use Mohr's circle for mechanical design, structural analysis, geotechnical engineering, and failure prediction. The reference table compares four failure criteria with their applicability to ductile and brittle materials.

When This Page Helps

Mohr's circle is a compact way to see how combined normal and shear stresses transform when the reference plane rotates. This calculator saves time on stress transformation, principal-stress checks, and failure criteria by showing the geometry and the derived values together.

It is useful in machine design, pressure vessels, slope stability, and any other combined-loading problem where the direction of the stress plane matters.

How to Use the Inputs

1. Enter σx, σy (normal stresses) and τxy (shear stress) in MPa.
2. Optionally enter the material yield strength for safety factor calculation.
3. Read principal stresses σ₁ and σ₂, maximum shear stress, and principal angle.
4. Enter a rotation angle θ to find stresses on a rotated plane.
5. Check Von Mises and Tresca stresses against the yield strength.
6. Use presets for common loading scenarios.

Example Calculation

Tips & Best Practices

• For pressure vessels, σx = PD/(2t) (hoop) and σy = PD/(4t) (axial) with τxy = 0.
• Maximum shear stress planes are always 45° from the principal planes.
• A hydrostatic state (σ₁ = σ₂) gives Mohr circle radius = 0 (pure compression or tension).
• In soil mechanics, Mohr-Coulomb failure is plotted with Mohr circles to find the critical friction angle.
• Remember that in Mohr's circle convention, positive τ may be plotted downward depending on the textbook.

How Mohr's Circle Works

The circle is built from the average normal stress at the center and the combined normal/shear radius. Rotating the stress element moves a point around the circle; the horizontal intercepts give the principal stresses and the top and bottom points give the maximum shear stress.

Interpreting the Failure Metrics

Von Mises is usually the preferred ductile-material check because it reflects distortion energy. Tresca is stricter and often gives a lower allowable limit. If your material is brittle, the principal-stress values themselves matter more than the shear criterion.

Practical Use

Mohr's circle is most valuable when the loading is not already aligned with a known axis. That is common in welded joints, shafts, retaining structures, and pressure-containing parts where stress direction changes with orientation.

Frequently Asked Questions

• What does Mohr's circle show?

  It graphically represents how normal and shear stresses change as the element is rotated. Every point on the circle corresponds to a different orientation of the stress element.

• When is Von Mises more appropriate than Tresca?

  Von Mises is better for ductile materials as it accounts for the distortion energy. Tresca is more conservative (gives a lower allowable stress) and is sometimes used in pressure vessel codes.

• What if τxy = 0?

  Then σx and σy are already the principal stresses and the principal angle is 0° (or 90°). The Mohr circle is centered on the σ-axis with no shear offset.

• Can I use this for 3D stress states?

  This calculator handles 2D plane stress. For full 3D analysis, you need three Mohr circles (one for each pair of principal stresses). The maximum shear stress in 3D may differ from the 2D result.

• What is the principal angle?

  The angle θp rotates the coordinate system to align with the principal directions, where shear stress is zero and normal stresses reach their extreme values.

• How does rotation angle differ from principal angle?

  The principal angle is fixed by the stress state. The rotation angle is user-chosen—enter any angle to see what stresses would exist on that plane.