Mohr's Circle Calculator

Mohr’s circle is one of those rare engineering tools that survived the calculator revolution because it explains itself geometrically faster than any formula sheet can. Christian Otto Mohr published the construction in 1882, and every mechanical and civil engineering curriculum has taught it ever since. The idea: take a stress state with normal stresses σx, σy and shear τxy, and rotate the coordinate frame. The transformed stresses trace out a circle in σ-τ space. Find the circle and you have every possible rotation at a glance.

The formulas:

σ_avg = (σx + σy) / 2 (center of the circle) R = √[((σx − σy)/2)² + τxy²] (radius) σ₁ = σ_avg + R (max principal stress) σ₂ = σ_avg − R (min principal stress) τ_max = R (absolute max in-plane shear stress) tan(2θp) = 2τxy / (σx − σy) (rotation to principal axes)

The principal stresses are the eigenvalues of the 2D stress tensor. Geometrically, they sit at the rightmost and leftmost points of the circle where shear stress is zero. Rotating the physical element by θp (note: half the angle on the circle) brings the stress state to its principal orientation.

Why this matters for design:

Most failure criteria are stated in terms of principal stresses, not the raw σx, σy, τxy of an arbitrary cut. To check whether a part will fail under a given loading, you find the principal stresses, then compare them to the material’s strength using a yield criterion:

• Tresca (maximum shear stress): yield when (σ₁ − σ₂)/2 ≥ σ_yield/2, i.e. R ≥ σ_yield/2. Conservative; assumes shear failure dominates.
• Von Mises (distortion energy): yield when √(σ₁² − σ₁σ₂ + σ₂²) ≥ σ_yield. Standard for ductile metals; matches experimental data better than Tresca for most steels.

For brittle materials like cast iron or concrete, the rules invert: max-principal-stress theory and Mohr-Coulomb theory are used because brittle failure is driven by tensile cracks, not shear flow.

Worked example, a shaft under combined load:

A 50 mm shaft carries a torque that produces τxy = 60 MPa shear stress on its surface, plus an axial compression that produces σx = -80 MPa (σy = 0 because there’s no transverse load).

σ_avg = (-80 + 0) / 2 = -40 MPa R = √[((-80 - 0)/2)² + 60²] = √(1600 + 3600) = √5200 = 72.1 MPa σ₁ = -40 + 72.1 = 32.1 MPa (tension) σ₂ = -40 - 72.1 = -112.1 MPa (compression) τ_max = 72.1 MPa

For a steel with σ_yield = 250 MPa, Tresca says yield when R = 125 MPa. We’re at 72.1, so factor of safety ≈ 1.7. Von Mises: σ_vm = √(32.1² + 112.1² + 32.1·112.1) ≈ 132 MPa, factor of safety ≈ 1.9.

Geometric reading of the circle:

• Plot point A at (σx, τxy) and point B at (σy, -τxy). The line connecting A and B is a diameter.
• The center of the circle is the midpoint at σ_avg on the σ-axis.
• The radius is the distance from center to either A or B.
• The rightmost point is σ₁; leftmost is σ₂.
• The top and bottom of the circle (where σ = σ_avg, |τ| = R) are the maximum-shear orientations.
• The angle from the diameter A-B to the σ-axis is 2θp on the circle, but θp in physical space.

That factor-of-two confusion is the single most common student error.

3D extension:

In 3D stress states, three Mohr’s circles are drawn simultaneously, one for each pair of principal planes (σ₁σ₂, σ₂σ₃, σ₁σ₃). The largest of these gives the absolute maximum shear stress in the body, which controls Tresca failure in 3D. Most introductory courses stop at 2D; advanced courses bring in 3D for problems like pressure vessels with axial load or torsion in addition to internal pressure.

Sign conventions matter:

Mechanical engineering and civil engineering have historically used different sign conventions for shear. The standard one used here: tension is positive normal stress, and positive shear on the positive face rotates the element counterclockwise. Always verify the convention in the textbook you’re using before applying any formula.

When Mohr’s circle does not help:

The construction is for stress, not strain. There is a separate “Mohr’s circle of strain” using ε and γ/2 (half the engineering shear strain) — same shape, different axes. Confusing the two in homework is a classic mistake.