Mohr's Circle Calculator — Principal Stresses, Von Mises & Tresca Safety Factors
The MechanixCalc Mohr's Circle calculator transforms any 2D stress state — combined normal and shear stresses on a plane element — into the principal stresses, the maximum shear stress, and the principal angle in one step. Enter σx, σy and τxy and the tool draws the circle, reads off σ1 and σ2, and computes the Von Mises and Tresca safety factors against the material yield strength. Optional 3D triaxial mode adds a third principal stress and reports the absolute maximum shear stress and the triaxiality ratio.
Built for structural, pressure-vessel and machine-design engineers who need a rapid, defensible stress check — and want to hand a reviewer a calculation that shows the governing equations, the failure-criterion envelope and a full worked result. Strain rosette back-calculation converts three gauge readings (0-45-90 or 0-60-120 layouts) directly to principal stresses and a Von Mises safety factor, while the multiaxial fatigue module applies Goodman, Gerber, Soderberg and Sines criteria to a cyclic 2D stress state.
What this calculator does
- Principal stresses (σ1, σ2) and principal angle θp via Mohr's circle — exact closed-form
- Von Mises (distortion-energy) and Tresca (max-shear-stress) safety factors against Sy
- 3D triaxial stress state with absolute max shear and triaxiality ratio (σm / σVM)
- Strain rosette back-calculation: 0-45-90 and 0-60-120 gauge layouts → principal stresses
- Multiaxial fatigue analysis: Goodman, Gerber, Soderberg, and Sines (1955) criteria
- Failure-envelope charts (Von Mises ellipse, Tresca hexagon, brittle-fracture Sut limit)
- Branded PDF engineering report with equations and full result table
Method & formulas
Stress transformation and Mohr's circle
For a 2D stress state defined by normal stresses σx and σy and shear stress τxy, the centre and radius of Mohr's circle are C = (σx + σy)/2 and R = √[((σx − σy)/2)² + τxy²]. The principal stresses are the intersections of the circle with the normal-stress axis; the maximum in-plane shear stress is the radius. The principal angle θp is half the angle on the circle from point A(σx, τxy) to the σ1 point, measured in the same rotational sense.
σ1, σ2 = (σx + σy)/2 ± √[((σx − σy)/2)² + τxy²]where σ1 = major principal stress (MPa); σ2 = minor principal stress (MPa); σx, σy = applied normal stresses (MPa); τxy = applied shear stress (MPa)
τmax = R = √[((σx − σy)/2)² + τxy²]; θp = ½ · atan2(2τxy, σx − σy)where τmax = maximum in-plane shear stress (MPa); θp = angle from x-axis to σ1 direction (degrees)
Von Mises and Tresca yield criteria
The Von Mises (distortion-energy) criterion predicts yielding when the equivalent stress σVM reaches the uniaxial yield strength Sy. For plane stress (σ3 = 0) this simplifies to the expression below in terms of σ1 and σ2. The Tresca (maximum-shear-stress) criterion predicts yielding when the largest shear stress across all planes reaches Sy/2. Tresca is more conservative in the biaxial compression quadrant; Von Mises gives a ~15% higher capacity along the shear diagonal. The calculator reports both and plots the operating point on a σ1–σ2 failure-envelope chart.
σVM = √(σ1² − σ1·σ2 + σ2²); SF_VM = Sy / σVMwhere σVM = Von Mises equivalent stress (MPa); σ1, σ2 = principal stresses (MPa); Sy = yield strength (MPa); SF_VM = Von Mises safety factor
τmax_abs = max(|σ1|, |σ2|, |σ1 − σ2|) / 2; SF_Tr = Sy / (2 · τmax_abs)where τmax_abs = absolute maximum shear stress across all planes (MPa); SF_Tr = Tresca safety factor
Multiaxial fatigue — Goodman and Sines criteria
When the stress state cycles between a mean and alternating component, the calculator computes Von Mises equivalent mean and alternating stresses (σm_VM, σa_VM) and evaluates the modified Goodman, Gerber and Soderberg criteria. The Sines (1955) criterion is also available: it uses the square root of the second deviatoric stress-invariant amplitude (√J2a) together with the mean hydrostatic stress to predict fatigue under fully general multiaxial cycling — a more physically motivated model than the scalar Goodman line for non-proportional loading.
σa_VM / Se + σm_VM / Su = 1 / SF_Goodmanwhere σa_VM = Von Mises alternating equivalent stress (MPa); σm_VM = Von Mises mean equivalent stress (MPa); Se = corrected endurance limit (MPa); Su = ultimate tensile strength (MPa); SF_Goodman = Goodman safety factor
√J2a + b · σm_hydro = Se / √3; b = Se / Suwhere √J2a = square root of second deviatoric stress-invariant amplitude; σm_hydro = mean hydrostatic stress = (σm_x + σm_y) / 3 (plane stress); Se = endurance limit (MPa); Su = ultimate tensile strength (MPa)
Worked example
A machine component is subjected to σx = 60 MPa, σy = 0 MPa, and τxy = 40 MPa. The material yield strength is Sy = 250 MPa. Find the principal stresses, the maximum shear stress, the principal angle, and both safety factors.
Given
- Normal stress σx60 MPa
- Normal stress σy0 MPa
- Shear stress τxy40 MPa
- Yield strength Sy250 MPa
Result
- Principal stress σ180 MPa
- Principal stress σ2−20 MPa
- Maximum shear stress τmax50 MPa
- Principal angle θp≈ 26.57°
- Von Mises safety factor SF_VM≈ 2.73
- Tresca safety factor SF_Tr2.50
- Compute the centre of Mohr's circle: C = (σx + σy)/2 = (60 + 0)/2 = 30 MPa.
- Compute the radius: R = √[((σx − σy)/2)² + τxy²] = √[(60/2)² + 40²] = √[30² + 40²] = √[900 + 1600] = √2500 = 50 MPa.
- Principal stresses: σ1 = C + R = 30 + 50 = 80 MPa; σ2 = C − R = 30 − 50 = −20 MPa.
- Maximum in-plane shear stress: τmax = R = 50 MPa.
- Principal angle: θp = ½ · atan2(2 × 40, 60 − 0) = ½ · atan2(80, 60) = ½ × 53.13° ≈ 26.57°.
- Von Mises equivalent stress: σVM = √(σ1² − σ1·σ2 + σ2²) = √(80² − 80 × (−20) + (−20)²) = √(6400 + 1600 + 400) = √8400 ≈ 91.65 MPa.
- Von Mises safety factor: SF_VM = Sy / σVM = 250 / 91.65 ≈ 2.73.
- Tresca check: max(|σ1|, |σ2|, |σ1 − σ2|)/2 = |80 − (−20)|/2 = 100/2 = 50 MPa; SF_Tr = Sy / (2 × 50) = 250 / 100 = 2.50.
This example uses the Pythagorean triple (3-4-5 scaled to 30-40-50) for clean round numbers. The Tresca criterion is the more conservative result here (SF 2.50 vs 2.73). This is an illustrative calculation — verify against your actual geometry, loads and material data.
Frequently asked questions
Which standard does this Mohr's Circle calculator use?
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. The governing equations are shown in full in the generated PDF report.
What stress inputs does the calculator accept?
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.
What is the difference between Von Mises and Tresca, and which should I use?
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.
Can it handle a 3D stress state?
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).
Is the Mohr's Circle calculator free?
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 part of a paid plan.
Related calculators
- Shaft Analysis (DIN 743)Shaft fatigue uses the same Von Mises / Goodman combination — link the stress state from Mohr's Circle into the shaft safety-factor check.
- Fatigue AnalysisDetailed S-N curve life prediction and mean-stress correction for cyclic loading — extends the Goodman line from Mohr's multiaxial fatigue panel.
- Beam DeflectionBending generates σx and τxy on the cross-section; feed the beam bending and shear stresses directly into Mohr's Circle.
- Torsion of SectionsTorsional shear stress τ is the primary input to Mohr's Circle for a shaft under pure torque — find the principal stresses and Von Mises stress.
- Weld DesignWeld throat stresses (normal + shear) are a classic Mohr's Circle application — verify the resultant principal stress against the fillet-weld allowable.
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