Principal stresses and Mohr circle

At any point in a loaded body, the stresses depend on the orientation of the plane you look at. A small square element with normal stresses σx, σy and shear τxy, if rotated, shows different values. There are always two perpendicular planes on which shear is zero; the normal stresses on those planes are the principal stresses σ₁ and σ₂, the largest and smallest normal stresses at that point. The maximum shear stress acts on planes at 45° to them and equals half their difference.

This matters because materials fail on particular planes. Concrete cracks perpendicular to the principal tension, which is why diagonal shear cracks appear near beam supports. Soils and rocks slip on planes where shear is high relative to normal stress. Welds and plates are checked against combined stress criteria built from principal values.

Mohr's circle is a graphical way to see all of this. Plot σ on the horizontal axis and τ on the vertical, mark the points (σx, τxy) and (σy, −τxy), and draw a circle through them with its centre on the σ axis. The circle's extreme left and right points are the principal stresses, its radius is the maximum shear, and rotating the element by θ moves round the circle by 2θ. Soil mechanics uses the same circle with failure envelopes in triaxial testing.