Skip to content
All explainers

Mohr’s circle, explained

In Hebrew: מעגל מוהר

Mohr’s circle turns a 2D stress transformation into geometry. One circle gives the principal stresses, the maximum shear stress and the stress on any plane through a point.

How to draw it

Plot normal stress σ on the horizontal axis and shear stress τ on the vertical one. Mark the point for the x face, (σ_x, τ_xy), and the point for the y face, (σ_y, −τ_xy). The line between them is a diameter. Its midpoint on the σ axis is the centre C, and half its length is the radius R.

Every point on the circle is the stress on some plane through the same material point. Rotating the element by θ moves you 2θ around the circle. That doubling is the whole trick, and the most common place to slip.

What you read off it

The two points where the circle crosses the σ axis are the principal stresses, σ₁ = C + R and σ₂ = C − R. On those planes the shear stress is zero. The top and bottom of the circle give the maximum in-plane shear stress, equal to R, on planes 45° from the principal ones, and those planes carry normal stress C.

Sign conventions differ between textbooks, in particular whether positive shear is plotted up or down. Pick one, write it at the top of the page, and the rotation direction follows from it.

The third dimension

A plane-stress element still has a third principal stress: zero, on the free surface. The absolute maximum shear stress is half the difference between the largest and smallest of all three. When σ₁ and σ₂ have opposite signs, that is just R. When they share a sign, the zero stress becomes the extreme and the true maximum shear is larger than R. Failure criteria for ductile metals care about the true one.

A worked number

Take σ_x = 80 MPa, σ_y = −20 MPa and τ_xy = 30 MPa. The centre is C = 30 MPa. The radius is R = √(50² + 30²) ≈ 58.3 MPa. So σ₁ ≈ 88.3 MPa and σ₂ ≈ −28.3 MPa, and since they have opposite signs the maximum shear is 58.3 MPa.

The principal angle comes from tan 2θ_p = 60/100, so 2θ_p ≈ 31° and θ_p ≈ 15.5° from the x axis.

Common mistakes

  • Rotating the element by the angle measured on the circle, instead of half of it.
  • Flipping the shear sign for one face and not the other, which puts both points on the same side.
  • Forgetting the zero third principal stress, and under-reporting the maximum shear when σ₁ and σ₂ have the same sign.

Mechanical engineering, explained