Table of Contents
What Is Mohr's Circle?
Mohr's circle is a graphical method for visualizing and calculating the state of stress at a point in a material when subjected to combined loading. Developed by German engineer Christian Otto Mohr in 1882, it maps the relationship between normal and shear stresses on planes at different orientations. The circle's center represents the average normal stress, and its radius represents the maximum shear stress. Principal stresses are found at the points where the circle crosses the horizontal axis (zero shear).
Mohr's circle is invaluable in structural engineering, mechanical design, and geotechnical engineering. It quickly reveals the maximum and minimum normal stresses, maximum shear stress, and the orientations of critical planes. This information is essential for failure analysis using criteria like von Mises, Tresca, or Mohr-Coulomb, determining whether a material will yield, fracture, or slip.
Stress Transformation Formulas
Circle Construction
| Element | Value | Meaning |
|---|---|---|
| Center | (σx+σy)/2 | Average normal stress |
| Radius | √[(σx-σy)²/4 + τ²] | Max shear stress |
| Right intercept | Center + R | Major principal stress σ1 |
| Left intercept | Center - R | Minor principal stress σ2 |
| Top of circle | (Center, R) | Max shear plane |
Frequently Asked Questions
What are principal stresses?
Principal stresses are the normal stresses on planes where shear stress is zero. Every stress state has two principal stresses in 2D (three in 3D). They represent the maximum and minimum normal stresses possible at that point. Principal stress directions are perpendicular to each other and are rotated from the original x-y axes by the principal angle θp.
Why is maximum shear stress important?
Maximum shear stress determines yielding in ductile materials according to the Tresca criterion: yielding occurs when τmax reaches the material's shear yield strength (σy/2). The Tresca criterion is more conservative than von Mises and is simpler to apply. In Mohr's circle, τmax equals the circle radius and occurs on planes at 45 degrees to the principal directions.
How does Mohr's circle relate to 3D stress?
In 3D, there are three principal stresses (σ1 ≥ σ2 ≥ σ3) and three Mohr's circles connecting each pair. The absolute maximum shear stress is (σ1 - σ3)/2, which is the radius of the largest circle. For plane stress (σz = 0), the third principal stress is zero, and the out-of-plane shear stress may exceed the in-plane maximum.