Principal Stresses and Mohr's Circle

Cut through a point on one plane and you get one pair of stresses; cut on another and you get a different pair. Nothing about the loading changed — only the question. So "the stress here is 80 MPa" is an incomplete statement.

A two-dimensional stress state takes three numbers: σx, σy and τxy. Everything on any other plane follows from those three.

The transformation equations are sinusoidal in 2θ, so a full cycle takes 180° of physical rotation. Where the direct stress reaches its maximum, the shear stress is exactly zero — those are the principal planes, and the values there are the principal stresses σ₁ and σ₂.

Squaring and adding the transformation equations eliminates the angle and leaves the equation of a circle. Every plane through the point maps to one point on it, and rotating the physical plane by θ moves you 2θ around the circle — which is why it closes after 180°.

Because the top of the circle is 90° around from the axis crossings, τ_max acts on planes 45° from the principal planes. That single geometric fact explains slip bands, torsional fractures and the 45° shear failure of a ductile tensile specimen.

Pure shear has zero direct stress on the reference planes, so the circle is centred on the origin and the principal stresses are ±τ. A twisted shaft therefore carries equal tension and compression on planes at 45°.

Brittle materials fail on the tension plane and crack in a 45° helix; ductile ones fail in shear and break flat across the section. Twisting a stick of chalk demonstrates the first in about a second.

Material data comes from a uniaxial test: one stress, one direction, one number. A real component is under a combined state produced by bending, torsion and axial load together. The bridging question — which combination is as severe as the tensile yield stress — is what a failure theory answers, and Mohr's circle supplies its inputs.

The 15% gap between Tresca and von Mises is not academic when the factor of safety is 1.5. And using a ductile theory on a brittle material — or the reverse — is a genuine design error, not a matter of preference.

Two-dimensional analysis assumes the third principal stress is zero. That is fine for a free surface, which is where the maximum stress usually is — but in a thick pressure vessel or a rolling contact, all three are significant and the two-dimensional circle understates the shear.