Mohr's Circle Calculator

About the Mohr's Circle Calculator

Enter the stresses on an element — σx, σy and τxy for plane stress — and get the principal stresses σ1 and σ2 with the direction of each, the maximum in-plane shear stress and the planes it acts on, the stresses on a plane turned to any angle θ, the absolute maximum shear (including the out-of-plane one), and the von Mises and Tresca equivalent stresses with factors of safety against a yield strength. Everything is drawn on an interactive Mohr's circle: move the angle and the rotated plane travels round it.

For a general stress state, switch to the 3D stress tensor: the six components give the three principal stresses (the eigenvalues of the tensor), their directions, the invariants, the maximum shear and the equivalent stresses, with all three Mohr's circles. Results are estimates from elasticity theory for checking and learning, not a professional design.

Without a pass, the free preview draws the circle of your own stresses with a watermark and no numbers on it, and shows the names of the table’s first rows, with the principal stresses, the working and every other figure hidden; the full result, Copy result, the CSV table and the PNG diagram come with a Pro pass.

Common uses

• Finding the principal stresses and the maximum shear at a point of a beam, shaft or plate from a strain-gauge rosette or a hand calculation
• Combining bending and torsion in a shaft (σx from bending, τxy from torque) and checking the von Mises stress
• Reading the stresses on a weld, a glued joint or a grain direction at a given angle
• Getting the principal stresses and directions of a full stress tensor from a finite-element result
• Checking Mohr’s circle homework, with the working and the diagram

Formulas used

• Centre and radius: σ_avg = (σx + σy)/2, R = √(((σx − σy)/2)² + τxy²)
• Principal stresses: σ1,2 = σ_avg ± R; direction of σ1: θp = ½ atan2(2τxy, σx − σy), counter-clockwise from x
• Maximum in-plane shear: τmax = R on the planes at θs = θp − 45° (and +45°), where the normal stress is σ_avg
• Any plane: σx′ = σ_avg + (σx − σy)/2·cos 2θ + τxy·sin 2θ; τx′y′ = −(σx − σy)/2·sin 2θ + τxy·cos 2θ; σy′ = σx + σy − σx′
• Absolute maximum shear (plane stress, σ3 = 0): half the spread of σ1, σ2 and 0
• von Mises: σ_vm = √(σ1² − σ1σ2 + σ2²) = √(σx² − σxσy + σy² + 3τxy²); Tresca: σmax − σmin
• 3D: the principal stresses are the eigenvalues of the symmetric stress tensor, the roots of σ³ − I1σ² + I2σ − I3 = 0; σ_vm = √(½[(σ1 − σ2)² + (σ2 − σ3)² + (σ3 − σ1)²]); τ_oct = (√2/3)σ_vm

Sign convention on the circle

Tension is positive. On the circle the shear stress is plotted positive downward, and the x face is the point X = (σx, τxy), the y face Y = (σy, −τxy). With this convention a counter-clockwise turn of the element by θ is a counter-clockwise turn of 2θ on the circle, as in Hibbeler’s Mechanics of Materials; it draws the same picture as Beer and Johnston’s rule that a shear stress tending to turn the element clockwise is plotted above the axis. Angles in the results are measured counter-clockwise from the x axis.

Which equivalent stress to use

For ductile metals, yielding is predicted when the von Mises (distortion-energy) stress reaches the tensile yield strength; the Tresca (maximum-shear-stress) criterion is slightly more conservative, by up to about 15 %. Brittle materials such as cast iron and concrete fail by other criteria (Coulomb–Mohr, modified Mohr), which compare σ1 and σ3 with the tensile and compressive strengths separately.

Sources

• Mohr, O., the paper that introduced the circle, Der Civilingenieur, volume 28
• Beer, Johnston, DeWolf and Mazurek, Mechanics of Materials, chapter 7 (transformations of stress)
• Budynas and Nisbett, Shigley's Mechanical Engineering Design, chapter 3 (Mohr's circle, three-dimensional stress) and chapter 5 (von Mises and Tresca)

Limitations

• Linear elasticity and a single point: the stresses you enter are assumed to be known; the calculator does not work them out from loads.
• Plane stress assumes no stress perpendicular to the plane (σz = τxz = τyz = 0); use the 3D tensor otherwise.
• von Mises and Tresca are yield criteria for ductile, isotropic materials. They do not apply to brittle materials, composites or fatigue, and they do not include stress concentrations.
• An estimate for learning and checking — not a professional design. Have a qualified engineer verify anything safety-critical.

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