Mohr's Circle

The problem Mohr was solving

Take an infinitesimal cube of material inside a loaded body and look at one face. The stress state on that face is described by a normal stress σ and a shear stress τ. Rotate the cube — same material point, different orientation — and σ and τ change. Engineers need to know two things: which orientation gives the maximum normal stress (because that drives brittle fracture), and which orientation gives the maximum shear stress (because that drives ductile yielding). Both questions are eigenvalue problems on the symmetric 2x2 stress tensor [[σ_x, τ_xy], [τ_xy, σ_y]].

You can answer them with algebra: solve a quadratic for the principal stresses, then take the larger of the two off-diagonal extremes for the maximum shear. Or you can do it with a ruler, a compass, and a piece of graph paper. The second method is faster, harder to mistake, and produces a picture that explains itself. Otto Mohr, a German civil engineer, published it in 1882 and called it the Spannungskreis — stress circle.

Where the circle comes from

Start with the 2D stress transformation equations. Rotate an element by θ; the stresses on the rotated face are

σ_n(θ) = σ_avg + ((σ_x − σ_y)/2)·cos 2θ + τ_xy·sin 2θ
τ(θ)   = −((σ_x − σ_y)/2)·sin 2θ + τ_xy·cos 2θ

where  σ_avg = (σ_x + σ_y)/2

Rearrange and square both:

(σ_n − σ_avg)² = [((σ_x − σ_y)/2)·cos 2θ + τ_xy·sin 2θ]²
τ²            = [−((σ_x − σ_y)/2)·sin 2θ + τ_xy·cos 2θ]²

Add the two. The cross-terms cancel (cos² + sin² = 1) and you are left with

(σ_n − σ_avg)² + τ² = ((σ_x − σ_y)/2)² + τ_xy²
                    = R²

This is the equation of a circle in the (σ, τ) plane with center (σ_avg, 0) and radius R. Every possible (σ, τ) state for every possible rotation of the element lies on that one circle. Plot the original face as point X = (σ_x, τ_xy) and the perpendicular face as Y = (σ_y, −τ_xy); the diameter of the circle is the line XY. The whole construction follows from one trigonometric identity.

How to draw it

1. Compute σ_avg = (σ_x + σ_y)/2. This is the center of the circle.
2. Compute R = √(((σ_x − σ_y)/2)² + τ_xy²). This is the radius.
3. Plot point X = (σ_x, τ_xy) and point Y = (σ_y, −τ_xy). The line XY is a diameter.
4. Draw the circle.
5. Principal stresses: σ_1 = σ_avg + R (right intersection with σ-axis), σ_2 = σ_avg − R (left intersection).
6. Maximum shear: τ_max = R, at the top and bottom of the circle.
7. To find stresses on a face rotated θ in physical space, rotate point X by 2θ on the circle.

That's all of it. Three numbers in, four numbers out, no algebra past addition and a square root.

Worked example: combined tension and shear

A thin-walled pressure vessel section sees σ_x = 80 MPa (hoop), σ_y = 40 MPa (axial), τ_xy = 30 MPa (shear from torsion). Find the principal stresses, the maximum shear, and the orientation of the principal planes.

Step 1 — center:
  σ_avg = (80 + 40)/2 = 60 MPa

Step 2 — radius:
  R = √(((80 − 40)/2)² + 30²)
    = √(20² + 30²)
    = √(400 + 900)
    = √1300
    ≈ 36.1 MPa

Step 3 — principal stresses:
  σ_1 = 60 + 36.1 = 96.1 MPa
  σ_2 = 60 − 36.1 = 23.9 MPa

Step 4 — maximum in-plane shear:
  τ_max = R = 36.1 MPa

Step 5 — principal angle:
  tan 2θ_p = 2τ_xy / (σ_x − σ_y)
           = 2·30 / (80 − 40)
           = 60 / 40 = 1.5
  2θ_p = 56.3°
  θ_p  = 28.15°

The major principal stress acts on a face rotated 28.15° from the x-face. The maximum shear plane is rotated another 45° from that, at 73.15° from x. Total time on a back-of-envelope: about 90 seconds. The circle costs you a sketch and gives you all four numbers at once.

Why the angles double

The transformation equations are functions of 2θ, not θ. Geometrically, this means the point on Mohr's circle representing a face rotated θ in physical space sits at angular position 2θ on the circle, measured from the original face's point. Rotate the element 90° (perpendicular face) and you walk 180° around the circle — to the diametrically opposite point, which makes sense because the perpendicular face sees σ_y where the original saw σ_x and the same shear with reversed sign.

The factor of two is a structural feature of symmetric second-rank tensors in 2D — it isn't an accident of Mohr's construction. It also has a deeper interpretation: a 180° rotation returns the element to its original state with axes relabelled, so one full trip around the circle corresponds to a 180° physical rotation, not 360°.

The strain circle

The same construction works for the strain tensor, with one bookkeeping change: plot (ε, γ/2) rather than (σ, τ), because the engineering shear strain γ is twice the tensor shear strain. The center sits at ε_avg = (ε_x + ε_y)/2; the radius is R = √(((ε_x − ε_y)/2)² + (γ_xy/2)²). Principal strains, maximum shear strain, and any rotated strain state follow by the same construction. This is what strain rosettes are reduced to in the field: three gauges, three normal strains, one Mohr's circle, two principal directions.

Three dimensions: three circles

A 3D stress state has three principal stresses σ_1 ≥ σ_2 ≥ σ_3 and three principal axes. The Mohr construction generalises naturally:

Circle	Center	Radius	Meaning
C₁₂ (in σ_1−σ_2 plane)	(σ_1 + σ_2)/2	(σ_1 − σ_2)/2	Stresses on planes containing the σ_3 axis
C₂₃ (in σ_2−σ_3 plane)	(σ_2 + σ_3)/2	(σ_2 − σ_3)/2	Stresses on planes containing the σ_1 axis
C₁₃ (in σ_1−σ_3 plane)	(σ_1 + σ_3)/2	(σ_1 − σ_3)/2	Outer envelope — contains the other two

All admissible (σ, τ) pairs for any plane through the point lie in the crescent region bounded externally by the outer circle C₁₃ and externally by the two smaller circles C₁₂ and C₂₃. The absolute maximum shear in 3D is the radius of the outer circle:

τ_abs,max = (σ_1 − σ_3) / 2

This number — not the in-plane maximum from the 2D circle — drives the Tresca yield criterion in three dimensions. A subtle point: a 2D plane-stress analysis with σ_1 = 100 MPa, σ_2 = 20 MPa would give τ_max = 40 MPa from the in-plane circle. But if σ_3 = 0 (free surface), the outer 3D circle uses (100 − 0)/2 = 50 MPa, and that is the controlling value. Forgetting σ_3 is a classic exam trap.

The pole method

There is a slick variant called the pole construction. On Mohr's circle, the pole is a single point with a useful property: a line drawn through the pole at any orientation intersects the circle at the (σ, τ) state for the plane with that same physical orientation. To find the pole, draw a line through the point representing one face (say, the x-face) parallel to that face in physical space; the second intersection of that line with the circle is the pole.

Once you have the pole, you can read off the stress on any inclined plane by drawing a line through the pole parallel to that plane. No 2θ computation, no protractor — just rulers and the geometry of the circle. Soil mechanics teaches the pole method first because slip planes in soils are always at angles you'd rather read off geometrically.

Failure criteria sit on the circle

• Tresca (maximum shear). Yield occurs when τ_max = σ_y / 2. On Mohr's circle, that's when the radius reaches half the uniaxial yield stress. In 3D, use the outer circle's radius.
• von Mises (distortional energy). Yield when σ_vm = √(½[(σ_1−σ_2)² + (σ_2−σ_3)² + (σ_3−σ_1)²]) = σ_y. The geometric interpretation on Mohr's circles is less clean than Tresca's, but the same principal stresses are the inputs.
• Mohr-Coulomb (brittle / soils). A linear failure envelope τ = c + σ·tan φ drawn tangent to the largest Mohr circle that fits. Used for rock, concrete, and soil; the angle φ is the friction angle.
• Maximum normal stress (Rankine). Brittle materials fail when σ_1 reaches σ_t (uniaxial tensile strength). Read σ_1 directly off the right edge of the circle.
• Hoek-Brown (rock mechanics). Non-linear envelope tangent to the σ_1−σ_3 circle; used for rock masses where Mohr-Coulomb is too coarse.

Where Mohr's circle still earns its keep

• PE / FE licensure exams. Both the Civil/Structural and Mechanical PE morning sessions reliably contain at least one problem solvable by sketching a Mohr's circle in 60 seconds. Multiple-choice grading rewards graphical estimation.
• Sanity-checking FEA. Pull a stress tensor from a worst-case Gauss point, draw the circle, confirm the principal stresses are physically reasonable. If von Mises in the FEA output disagrees with the algebra implied by the circle, the boundary conditions are usually wrong.
• Strain rosette reduction. Three resistance strain gauges at known angles give three normal strains. The strain Mohr's circle backs out principal strains and directions in two steps.
• Soil mechanics. The Mohr-Coulomb failure envelope is drawn directly on top of a Mohr's circle for the soil's stress state at depth. The slope-stability analysis a geotechnical engineer does on day one of practice is still a Mohr's circle question.
• Welded joints and connections. The combined-stress check on a fillet weld under axial + shear loading is a Mohr's circle with τ_max compared to the weld's allowable shear stress.
• Teaching the eigenvalue structure of symmetric tensors. Every undergraduate solid mechanics course uses Mohr's circle as the cheapest visual for what an eigendecomposition means physically.

Common pitfalls

• Sign convention drift. Different textbooks plot τ positive downward (Mohr's original) or upward (most modern Western texts). Pick one and stick with it. The direction of rotation on the circle flips with the convention.
• Forgetting σ_3 in plane stress. Free surfaces have σ_3 = 0, which sometimes makes the outer 3D circle larger than the in-plane circle. Always check whether the in-plane τ_max or the (σ_1 − σ_3)/2 value controls.
• Doubling the angle twice. The factor of two is between physical and circle space — but rotation-angle formulas like tan 2θ_p already contain the 2θ. Don't multiply by two a second time.
• Misreading the strain circle. The strain Mohr's circle plots (ε, γ/2), not (ε, γ). Forgetting the factor of two on the shear axis gives principal strains that are wrong by ~14% and principal directions that are slightly off.
• Plane strain vs plane stress. In plane stress σ_z = 0; in plane strain ε_z = 0 (and σ_z ≠ 0). Both reduce to a 2D analysis but the through-thickness σ_z changes which 3D Mohr circle is largest.
• Applying it past yield. Mohr's circle is a linear-elastic transformation. Once any part of the material yields, the principal-stress directions can rotate with continued loading and the circle no longer represents the loaded state — only the elastic predictor of it.

Otto Mohr and the precursors

Otto Mohr (1835-1918) was a structural engineer and professor at Dresden Polytechnic. He published the stress circle in 1882 in Civilingenieur, building on an earlier 1866 graphical method by Karl Culmann and a parallel construction by William Rankine in Britain. Mohr's contribution was the clean (σ, τ) plotting convention and the demonstration that the construction generalises to strain and to three dimensions. He also developed the Mohr-Coulomb failure envelope — drawn tangent to the largest stress circle that the material can sustain — which remains the workhorse failure criterion for rock and soil. The circle is the single most-taught artifact in the history of solid mechanics curricula; only the free-body diagram has been drawn more times.