Christian Otto Mohr and the Oblique Plane Problem
By the 1880s, engineers could calculate normal and shear stresses in beams. But which combination on which plane through a material point was most likely to cause failure? Finding this required rotating the stress state to a new reference frame — two simultaneous equations with limited geometric insight. Mohr recognised that the transformation equations for plane stress have the mathematical form of a circle when plotted in stress space. Rather than solving equations, you draw a circle, and the answers are visible geometrically. The method was instantly adopted and has remained in every mechanics of materials curriculum since.
Constructing Mohr's Circle
As you rotate the plane by angle θ in physical space, the representative point moves by 2θ around Mohr's circle. Every point on the circle is a valid stress state on some plane through the point.
The Von Mises Criterion: The Standard for Ductile Failure
Von Mises is physically grounded in the observation that hydrostatic stress does not cause yielding in metals — only the distortional component matters. Experiments confirm that von Mises predicts the onset of yielding more accurately than the simpler Tresca criterion, making it the default choice in mechanical design and FEM post-processing.
Three-Dimensional Stress States
In three dimensions, a stress state has six independent components and the Mohr's circle construction produces three circles — one for each principal stress pair. The absolute maximum shear stress (the largest radius) governs failure in ductile materials under triaxial loading. Three-dimensional principal stress analysis is important in pressure vessel design, shaft design under combined loading, and geotechnical analysis of soil stress states. FEM tools routinely display von Mises stress as colour contour plots — but the engineer must understand what those colours mean and which criterion is appropriate. Enter σx, σy, and τxy. EngForge draws the complete Mohr's circle, reads off principal stresses, maximum shear stress, and principal plane orientation, then computes von Mises stress and compares it to your material's yield strength.