Mohr's circle

The formula

Start with the plane-stress transformation equations. Cut a stressed element on a plane rotated θ from the x-axis and the stresses on that cut are:

σ_n = (σx + σy)/2 + ((σx − σy)/2)·cos 2θ + τxy·sin 2θ — the normal stress on the rotated plane: an average part that never changes, plus an oscillating part that swings with twice the rotation angle.

τ_n = −((σx − σy)/2)·sin 2θ + τxy·cos 2θ — the shear stress on the same plane, ninety degrees out of phase with the normal part.

Square both, add them, and the θ terms cancel. What's left is a circle in the (σ, τ) plane:

(σ_n − C)² + τ_n² = R² — every plane you can cut lands on this circle; nothing lands off it.

C = (σx + σy)/2 — the center: the average normal stress, an invariant of the rotation.

R = √( ((σx − σy)/2)² + τxy² ) — the radius: how far the stress state can swing away from that average.

The numbers engineers actually pull off the circle:

σ1 = C + R and σ2 = C − R — the principal stresses, where the circle crosses the σ axis and shear is exactly zero.

τ_max = R — the maximum in-plane shear, at the top of the circle, on planes 45° from the principal planes.

tan 2θp = 2·τxy / (σx − σy) — the rotation from your axes to the principal axes, remembering the circle works in double angles.

For strain, the same circle works with one substitution: plot ε against γ/2, half the engineering shear strain, and every relation above carries over.

Where you meet it

Data reduction on a test stand. A 0°/45°/90° strain rosette gives three numbers; Mohr's circle of strain turns them into principal strains and their direction. When the principal direction comes out 30° off the axis everyone assumed was the load path, the circle is how you find out — and the rework it triggers is why rosettes exist.
Shaft sizing anywhere torque and bending meet: a pump shaft, a gearbox input, an actuator. Bending gives σ on the axial face, torsion gives τ, and the circle combines them into the σ1 and τ_max your fatigue and yield checks actually need.
A failure review board holding a broken part. Ductile shafts in torsion shear off flat; brittle ones — chalk, cast iron, over-hard steel — crack along a 45° helix, because pure shear puts the principal tension at 45° to the axis. The circle is the argument for which failure mode you're looking at.
Geotech and materials labs. Triaxial soil tests are reported as a family of Mohr circles, and the line drawn tangent to the failed ones is the Mohr–Coulomb envelope that sets bearing capacity and slope stability numbers.

How it works

The circle is not a model or an approximation. It is the tensor transformation law for stress at a point, drawn instead of computed — exact for any material, elastic or not, as long as the continuum assumption holds at that point. What you assume about the material comes later, when you decide what σ1 or τ_max means for failure.

The one geometric rule that runs everything: a rotation of θ on the part is a rotation of 2θ around the circle. Faces 90° apart on the element sit 180° apart on the circle — diametrically opposite, which is why the x-face point and the y-face point define a diameter. Forget the factor of two and every angle you report is wrong by half.

A worked pass, numbers checked. Take σx = 80 MPa, σy = 20 MPa, τxy = 40 MPa:

C  = (80 + 20)/2 = 50 MPa
R  = √( ((80−20)/2)² + 40² ) = √(30² + 40²) = 50 MPa
σ1 = 50 + 50 = 100 MPa      σ2 = 50 − 50 = 0 MPa
τ_max (in-plane) = 50 MPa
2θp = atan(2·40 / (80−20)) = 53.13°  →  θp = 26.6°

Rotate the element 26.6° and the shear vanishes; all the load shows up as 100 MPa of pure tension on one face. Same point, same physics, different cut.

The gotchas, in the order they bite:

The third circle. Plane stress still has a third principal stress, σ3 = 0, and the full 3-D picture is three circles. The absolute maximum shear is always (σ_max − σ_min)/2 taken over all three principal stresses, zero included. Both in-plane stresses tensile: the out-of-plane circle governs and τ_abs = σ1/2. Both compressive: it's the other out-of-plane circle and τ_abs = |σ2|/2 — the case a triaxial soil test lives in. Opposite signs: zero sits between them and the in-plane (σ1 − σ2)/2 already governs. A thin-wall vessel at 200 MPa hoop and 100 MPa axial has an in-plane τ_max of 50 MPa but an absolute τ_max of 100 MPa — miss the third circle and your shear-based margin is optimistic by a factor of two.
The strain circle uses γ/2. Engineering shear strain is twice the tensor component, and the circle is a tensor object. Plot raw γ from a rosette reduction and the radius, the principal angle, and everything downstream is wrong. This is the single most common Mohr's-circle error in test data reduction.
Sign conventions are not standardized. Mechanics-of-materials texts usually plot shear positive downward (or count clockwise-rotating shear as positive) so that physical rotations and circle rotations go the same direction; geomechanics flips signs so compression is positive. The circle is agnostic. Pick one convention, write it on the plot, and don't mix sources mid-calculation.
The circle describes one point. A beam's flange and web live at different places on different circles. Drawing one circle for "the part" is meaningless; the circle is evaluated wherever your stresses were, and nowhere else.
It answers rotation, not failure. Whether 100 MPa of principal tension breaks anything depends on the material and the criterion — max normal stress for brittle parts, τ_max or von Mises for ductile ones. Von Mises, note, cannot be read directly off a single circle; it mixes all three principal stresses.

The mistake people make most is treating the circle as a relic that software replaced. The FEA post-processor is running the same transformation law; the circle is how you check it, and how you catch the model whose "max stress" is a σx component nobody rotated.

History

The circle carries the wrong man's name, which its own literature freely admits. Karl Culmann — chair of engineering science at the Zurich Polytechnic from 1855, author of Die graphische Statik (1865), and the driving force behind the graphical-methods school of nineteenth-century engineering — was the first to represent beam stresses as a circle, while working out how longitudinal and vertical stresses combine in a bent girder ^[1]^[2]^[4].

Christian Otto Mohr took it the rest of the way. Born in 1835 in Wesselburen, Holstein, Mohr spent his early career from 1855 as a railway engineer for the Hanover and Oldenburg state lines, designing bridges and early steel trusses, before taking professorships at the Stuttgart Polytechnic in 1867 and Dresden in 1873 ^[2]^[3]. In 1882, in the journal Der Civilingenieur, he published the paper that made the circle a general tool — extending the construction from Culmann's beam case to arbitrary two- and three-dimensional stress states, and to strain ^[1]^[2]^[3]^[6]. This was a slide-rule world; a construction that turned tensor rotation into compass-and-straightedge work was not a novelty, it was a productivity tool ^[1]^[3].

Mohr then used his own diagram to attack the failure question, proposing a strength theory built on the stress circle at a time when most of the field followed Saint-Venant's maximum-strain criterion ^[1]^[2]. Drawn as an envelope tangent to the circles at failure, his theory generalized Coulomb's 1776 friction-and-cohesion work into what soil and rock mechanics still run on as the Mohr–Coulomb criterion ^[5]. He kept working in Dresden past his 1900 retirement until his death in 1918 ^[2]^[3].

Related tools

/tools/stress-strain — the σ and ε inputs the circle rotates
/tools/strain-gauge-bridge — where the rosette numbers come from before the strain circle reduces them
/tools/torsion-shaft — the pure-shear point that sits centered on the origin of the circle
/tools/pressure-vessel-hoop — the classic biaxial state, and the out-of-plane τ_max trap
/tools/beam-deflection — the bending stresses Culmann was drawing circles for in 1865
/tools/section-modulus-rect — turns moment into the σx that anchors the circle
/tools/weld-throat-stress — combined normal and shear on a throat plane, a Mohr problem in work clothes

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →