Real parts rarely experience a single, simple stress — they carry combinations of tension, compression, and shear acting in multiple directions at once.
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Walk-through
How to Use This Calculator
3 steps▸
1
Pick the tab that matches your data
Use the From Principal tab when you already know the three principal stresses (σ1, σ2, σ3) — for example, from a Mohr's circle analysis. Use the From Components tab when you have a general 3D stress state expressed as normal stresses (σx, σy, σz) and shear stresses (τxy, τyz, τzx) directly off an FEA model or a raw stress tensor. Use the Yield Check tab when you want a safe/yields verdict against a specific material's yield strength.
2
Enter stresses in matching units
All stress inputs are entered in megapascals (MPa) by default. For a 2D (plane-stress) problem, set the out-of-plane stress (σ3 on the principal tab, or σz/τyz/τzx on the components tab) to zero. Compressive stresses can be entered as negative numbers — the von Mises formula only depends on the differences between stresses, so the sign convention doesn't change the result's validity.
3
Read the equivalent stress and, if checking yield, the verdict
The result card shows the von Mises equivalent stress σv — a single scalar that combines the multi-axial stress state into one number. On the Yield Check tab, the interpretation line and the reference-material table below the inputs show whether σv is under or over the yield strength you entered, plus the resulting factor of safety.
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Reference
Formula & Methodology
2 formulas▸
From Principal Stresses
σv = √(½[(σ1−σ2)² + (σ2−σ3)² + (σ3−σ1)²])
The von Mises equivalent stress from three principal stresses σ1, σ2, and σ3 (all in the same units, typically MPa). This comes from the distortion-energy theory of yielding: a material yields when the energy associated with shape change (as opposed to volume change) reaches a critical value. Example: σ1 = 150 MPa, σ2 = 50 MPa, σ3 = 0 MPa → σv = √(½[(100)² + (50)² + (−150)²]) = √17,500 ≈ 132.29 MPa. In the uniaxial special case where σ2 = σ3 = 0, this reduces exactly to σv = σ1.
The equivalent form for a general 3D stress state expressed as normal stresses (σx, σy, σz) and shear stresses (τxy, τyz, τzx) — the form most FEA software reports directly, avoiding the need to first solve for principal stresses. Example: σx = 100 MPa, σy = 50 MPa, σz = 0 MPa, τxy = 30 MPa, τyz = 0, τzx = 0 → σv = √(7,500 + 2,700) ≈ 100.99 MPa. This form and the principal-stress form always agree for the same underlying stress state, since the principal stresses are just an eigenvalue rotation of the component stresses.
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Glossary
Key Terms Explained
7 terms▸
Von Mises Stress ↗A single equivalent scalar stress (σv) derived from a multi-axial stress state, used to predict yielding in ductile materials under complex loading. Also called the equivalent tensile stress.
Equivalent Stress ↗Any scalar value computed from a full multi-axial stress state that can be compared directly against a material's uniaxial strength (from a simple tension test) to predict failure. Von Mises stress is the most widely used equivalent stress for ductile metals.
Distortion Energy ↗The portion of a material's total strain energy associated with changing shape (as opposed to changing volume). The von Mises criterion states that yielding begins when distortion energy per unit volume reaches the same value it has at yield in a simple uniaxial tension test.
Yield Criterion ↗A mathematical rule that predicts when a material transitions from elastic to plastic (permanent) deformation under a general, multi-axial stress state. Von Mises is the most common criterion for ductile metals; Tresca (maximum shear stress) is a more conservative alternative.
Principal Stress ↗One of three mutually perpendicular normal stresses (σ1 ≥ σ2 ≥ σ3) that fully describe a 3D stress state with zero shear on the planes they act on. Any general stress state can be rotated into its principal-stress form.
Yielding ↗The onset of permanent (plastic) deformation in a material, occurring once the applied stress state exceeds the material's yield strength as measured against the chosen yield criterion.
Factor of Safety ↗The ratio of a material's yield strength to the applied (von Mises) equivalent stress: FoS = yield strength / σv. A factor of safety greater than 1 means the design has margin before yielding; less than 1 means it has already yielded.
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Scenarios
Real-World Examples
3 worked examples▸
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Uniaxial Tension Bar
Sanity check: von Mises reduces to the applied stress
σv = √(½[(100)² + (0)² + (−100)²]) = √10,000 = 100 MPa — exactly equal to σ1. This is the standard sanity check for the formula: under pure uniaxial tension, the von Mises equivalent stress always equals the applied stress, so a simple tension test directly measures the von Mises yield strength.
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Biaxial Bracket Under Combined Loading
Two principal stresses combine into one equivalent number
σv = √(½[(100)² + (50)² + (−150)²]) = √17,500 ≈ 132.29 MPa. Notice this is less than the simple sum (150 + 50 = 200 MPa) but more than the largest single stress (150 MPa) — the von Mises formula captures how the two principal stresses interact rather than treating them independently.
σv ≈ 132.29 MPa, well under the 250 MPa yield strength of mild steel — factor of safety = 250 / 132.29 ≈ 1.89. The bracket is safe against yielding under this load, with roughly 89% additional stress margin before the material begins to deform permanently.
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Reference
Cite This Calculator
APA & MLA▸
Use either format to cite this calculator in a paper, report, or resource list.
Real parts rarely experience a single, simple stress — they carry combinations of tension, compression, and shear acting in multiple directions at once. Von Mises stress collapses that entire multi-axial stress state into one number that engineers can compare directly against a material's yield strength from a basic tension test, making it the default failure check for ductile metals in everything from brackets to pressure vessels.
Why Combine Stresses Into One Number
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A material's yield strength is normally measured with a simple uniaxial tension test — pull a sample until it starts to deform permanently, and record that stress. But real components are rarely loaded in pure tension; they see a mix of normal and shear stresses acting on multiple planes simultaneously. Von Mises stress bridges this gap by computing a single equivalent scalar from the full multi-axial stress state that behaves, for yielding purposes, exactly like a uniaxial stress. That's what makes the direct comparison to a tension-test yield strength valid.
Distortion Energy vs. Total Strain Energy
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The theoretical basis for the von Mises criterion is that a material yields when the distortion energy per unit volume — the strain energy associated with changing shape — reaches the same critical value it has at yield in a simple tension test. Total strain energy is deliberately excluded: hydrostatic (equal-in-all-directions) stress changes a material's volume but not its shape, and ductile metals can withstand very high hydrostatic pressure without yielding. That's why the von Mises formula depends only on the differences between principal stresses, not their absolute magnitudes — a stress state that is uniformly shifted up or down by the same hydrostatic amount produces the identical σv.
Von Mises vs. Tresca
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The Tresca (maximum shear stress) criterion is the main alternative, defining yielding as the largest principal stress difference (σ1 − σ3) reaching the uniaxial yield strength. Tresca is more conservative — it predicts yielding at a lower equivalent stress than von Mises for the same stress state, with the two criteria agreeing exactly only for pure uniaxial or pure shear states. Von Mises is more widely used because it matches experimental yield data for ductile metals more closely and produces a smoother (rather than faceted) yield surface, but Tresca remains common where a conservative margin is specifically wanted.
Limits and Edge Cases
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This calculator computes a static von Mises equivalent stress from the stress state you provide — it does not account for fatigue (repeated sub-yield loading can still cause failure over many cycles), stress concentrations (holes, notches, and fillets locally raise stress well above the nominal value used here), or brittle materials (von Mises applies to ductile yielding; brittle fracture is governed by different criteria, such as maximum normal stress). It also assumes you have already correctly resolved the stress state — from an FEA result, a hand calculation, or a Mohr's circle analysis — since the calculator itself does not derive stresses from loads and geometry.
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Questions
Frequently Asked Questions
6 questions▸
What is the formula for von Mises stress?+
From principal stresses: σv = √(½[(σ1−σ2)² + (σ2−σ3)² + (σ3−σ1)²]). From general stress components: σv = √(½[(σx−σy)²+(σy−σz)²+(σz−σx)²] + 3(τxy²+τyz²+τzx²)). Both forms describe the same distortion-energy-based equivalent stress and always agree for the same underlying stress state.
When does a material yield according to von Mises?+
A material yields when the computed von Mises equivalent stress σv exceeds its yield strength (from a simple uniaxial tension test), measured in the same units. The ratio yield strength / σv gives the factor of safety against yielding.
What happens under pure uniaxial stress?+
Under pure uniaxial tension (σ2 = σ3 = 0), the von Mises formula reduces exactly to σv = σ1 — the equivalent stress equals the applied stress. This is the defining sanity check: a simple tension test directly measures the material's von Mises yield strength.
Should I use the 2D or 3D form?+
Both tabs handle 2D and 3D — just set the out-of-plane terms to zero for a 2D (plane-stress) problem: σ3 = 0 on the principal tab, or σz = τyz = τzx = 0 on the components tab. The full 3D formula naturally covers the 2D case as a special condition.
What units does this calculator use?+
All stresses (principal, component, and yield strength) are entered in megapascals (MPa) by default. As long as you're consistent, any pressure/stress unit works the same way — the formula is a ratio-free combination of stresses in matching units.
How does von Mises compare to the Tresca criterion?+
Von Mises is less conservative than Tresca (maximum shear stress) — it predicts yielding at a somewhat higher equivalent stress for the same multi-axial state, except in pure uniaxial or pure shear cases where the two agree exactly. Von Mises is the more commonly used criterion for ductile metals because it matches experimental yield data more closely.
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