Mohr's Circle Calculator — Principal Stresses, Von Mises & Tresca Safety Factors

Mohr's Circle is classical stress-transformation theory from the mechanics of materials, not governed by a single numbered design standard. The failure criteria are the Von Mises distortion-energy theory and the Tresca maximum-shear-stress theory — both underpin pressure-vessel codes (ASME BPVC) and structural design standards. The multiaxial fatigue module applies the modified Goodman, Gerber, Soderberg and Sines (1955) criteria. Important: this calculator compares the equivalent stress directly against the yield strength Sy — it is NOT an ASME BPVC code check. There is no stress linearization, no Pm/PL/Pb/Q stress categorization and no code allowable (1.5S/3S) applied, so it does not substitute for a Division 1 or Division 2 assessment. The governing equations are shown in full in the generated PDF report.

Enter the three components of the in-plane stress tensor: normal stress σx (on the x-face), normal stress σy (on the y-face), and shear stress τxy. Positive shear follows the engineering convention (positive τxy acts upward on the positive-x face). For 3D triaxial analysis, add a third principal stress σz. The strain rosette module takes three microstrain gauge readings and back-calculates the full stress tensor.

Both criteria predict the onset of yielding in a ductile material. Von Mises (distortion-energy) uses the full second deviatoric invariant and matches experimental yield data slightly better for most metals, allowing up to ~15% more capacity along the pure-shear diagonal. Tresca (maximum-shear-stress) is more conservative — it is the basis of many pressure-vessel codes (ASME BPVC Sec. III / VIII). For a conservative design code check, use Tresca; for best-estimate capacity of ductile steel, Von Mises is preferred. Both results are shown on the same failure-envelope chart.

Yes. Enable the 3D triaxial mode and enter the out-of-plane principal stress σz. The calculator re-orders σ1 ≥ σ2 ≥ σ3, computes the absolute maximum shear stress across all three principal planes, recalculates the Von Mises equivalent from all three principals, and reports the stress triaxiality ratio (hydrostatic stress divided by Von Mises stress — relevant for fracture-mechanics assessments).

The headline reports the GOVERNING safety factor — the lower of the Von Mises and Tresca values, which in practice means Tresca whenever the two differ, since the maximum-shear criterion is never the less conservative of the two. The bands are SAFE at SF ≥ 1.5, CHECK between 1.0 and 1.5 (below yield, but with less margin than most design practice allows), and FAIL below 1.0, where the material is predicted to yield. Every surface uses these same bands — the hero, the per-criterion pills, the stress-element sketch and the downloadable PDF report — so the report you hand to a reviewer can never read greener than the screen. The multiaxial fatigue panels are banded more strictly (SAFE at SF ≥ 2.0) because an infinite-life fatigue margin is a different quantity from a static yield margin.

The calculation is refused and the tool says why, rather than answering. A blank ultimate or yield strength, a zero or negative Young's modulus, a Poisson ratio outside the physical range −1 < ν < 0.5 (a common slip is entering 30 instead of 0.30), an unrecognised material key or an unrecognised strain-rosette layout all return a named error, a page-wide alert, and paused exports. This matters because those inputs sit in the denominator of a safety factor: before this behaviour a blank strength field produced the greenest reading the tool could express. A safety factor is only shown when it was actually computed — a dash means no result, not a large margin.

You can use it during a free 30-minute preview with no sign-up, and a free 14-day account trial unlocks every calculator with no credit card required. The branded PDF engineering report and saved calculations are included in the free 14-day trial and in every paid plan.