Von Mises stress is a single value, calculated from all the stress components at a point, that predicts whether a ductile material will yield. It is defined so that yield begins when the von Mises stress reaches the material’s yield strength from a simple tensile test, whatever mix of tension, compression and shear produced it. That is why it is the default result on most FEA stress plots.
It works well for ductile metals such as steel, aluminium and titanium under static load. On its own it is the wrong measure for brittle materials such as cast iron, glass and ceramics, for fatigue, and for composites, where other criteria apply.
Why a single equivalent stress is needed
At any point inside a loaded part, the stress state has six independent components: three direct stresses, σx, σy and σz, and three shear stresses, τxy, τyz and τzx. Material strength, on the other hand, usually comes from a tensile test, which produces one number: the yield strength, written here as Sy. To compare the two, the six components have to be reduced to one equivalent value.
For any stress state there is an orientation in which all the shear stresses vanish. The direct stresses on those planes are the principal stresses, written σ₁ ≥ σ₂ ≥ σ₃. They are the starting point for every failure criterion.
The von Mises formula
In terms of the principal stresses:
σvm = √( ½ [ (σ₁ − σ₂)² + (σ₂ − σ₃)² + (σ₃ − σ₁)² ] )
In terms of the six components in any coordinate system:
σvm = √( ½ [ (σx − σy)² + (σy − σz)² + (σz − σx)² ] + 3 (τxy² + τyz² + τzx²) )
For plane stress, as in a thin plate or at the free surface of a shaft, where σz, τyz and τzx are zero:
σvm = √( σx² − σxσy + σy² + 3τxy² )
Yield is predicted when σvm reaches Sy. Three special cases show how the criterion behaves:
- Uniaxial tension. With only σ₁ = σ acting, the formula returns σvm = σ, so the criterion matches the tensile test exactly.
- Pure shear. With σ₁ = τ, σ₂ = 0 and σ₃ = −τ, it gives σvm = √3 × τ. Yield in shear is therefore predicted at τ = Sy / √3, about 0.577 Sy.
- Equal triaxial stress. With σ₁ = σ₂ = σ₃, every difference is zero and σvm = 0. Hydrostatic pressure alone does not make a metal yield, which matches experiment.
What it represents physically
Von Mises is also called the distortion energy criterion. The strain energy stored in a stressed material can be split into a part that changes its volume and a part that changes its shape. Metals yield by slip along crystal planes, a shearing process driven by the shape-changing part. Von Mises stress measures that part and predicts yield when it reaches the value it has at yield in a tensile test. It is also proportional to the octahedral shear stress, which leads to the same criterion from a different starting point. Test results for most ductile metals under combined loading lie close to the von Mises prediction.
A worked example
A steel shaft carries a bending stress of 120 MPa and a torsional shear stress of 60 MPa at its surface. This is plane stress with σx = 120 MPa, σy = 0 and τxy = 60 MPa.
σvm = √(120² + 3 × 60²) = √25,200 ≈ 159 MPa
The principal stresses are σ₁ ≈ 145 MPa, σ₂ = 0 and σ₃ ≈ −25 MPa. The Tresca equivalent stress is σ₁ − σ₃ ≈ 170 MPa, and the maximum principal stress is 145 MPa. For a steel with a yield strength of 355 MPa, the factor of safety on yield is about 2.2 by von Mises and about 2.1 by Tresca. The two criteria differ by about 7% here. In pure torsion the gap would be at its widest, about 15%.
A second case shows that combined stresses can also lower the equivalent value. In a thin-walled cylinder under internal pressure p, with radius r and wall thickness t, the hoop stress is pr/t and the axial stress is pr/2t, while the radial stress is close to zero. The von Mises stress is then √0.75 × pr/t, about 0.87 times the hoop stress, whereas Tresca equals the full hoop stress. At 1 MPa (10 bar) in a 200 mm diameter tube with a 2 mm wall, that is a hoop stress of 50 MPa, an axial stress of 25 MPa and a von Mises stress of about 43 MPa. The axial tension makes the material slightly harder to yield than the hoop stress alone would suggest.
Von Mises, Tresca and maximum principal stress
| Criterion | Equivalent stress | Best suited to | Notes |
|---|---|---|---|
| Von Mises (distortion energy) | σvm as above | Ductile metals under static load | Default FEA output; always positive; closest to test data for most ductile metals |
| Tresca (maximum shear stress) | σ₁ − σ₃ | Ductile metals where a more conservative answer is wanted | Up to about 15% higher than von Mises, most of all in pure shear; equal to twice the maximum shear stress, and called stress intensity in some pressure vessel codes |
| Maximum principal stress (Rankine) | σ₁ in tension, σ₃ in compression | Brittle materials: grey cast iron, glass, ceramics | Sign matters, because tension opens cracks and compression closes them |
For materials much stronger in compression than in tension, criteria such as Mohr-Coulomb extend the principal stress approach by treating the two strengths separately.
When not to use von Mises stress
Brittle materials
Grey cast iron, glass and ceramics fail by fracture with little or no yielding, and are typically several times stronger in compression than in tension. Von Mises has no sign, so it cannot tell a tensile region from a compressive one. A compressive region of a cast iron housing can show a high von Mises value and be quite safe, while a lower value in tension elsewhere is the real risk. Use maximum principal stress, or a criterion that treats tension and compression separately.
Fatigue
Because von Mises stress is always positive, it loses the direction of the load. A uniaxial stress alternating between +100 MPa and −100 MPa has a true range of 200 MPa, yet its von Mises value simply rises from zero to 100 MPa and back twice in every cycle. Fatigue cracks are driven by stress range, particularly the principal stress acting across the crack, so fatigue assessment uses signed equivalent stresses, principal stress ranges or critical plane methods, and welded joints are usually assessed on the stress range across the weld toe. Our fatigue analysis guide explains these in more depth.
Composites and other directional materials
Carbon and glass fibre laminates are many times stronger along the fibres than across them, and fail in distinct ways: fibre breakage, matrix cracking and delamination. Von Mises assumes the same strength in every direction. Ply-by-ply criteria such as maximum stress, maximum strain, Tsai-Wu and Hashin are used instead. Wood and many 3D-printed parts are also direction-dependent.
Polymers, rubbers and stability
Many polymers yield at a higher stress in compression than in tension, so von Mises is only an approximation for them. Rubbers are better judged on strain. And a low von Mises stress says nothing about stability: thin walls and slender struts can buckle at stresses well below yield.
Reading a von Mises plot
Compare the peak with the yield strength at the operating temperature and in the condition actually used: welded, cast or heat-affected material can be much weaker than handbook figures for wrought parent metal. The ratio of the two is the factor of safety on yield, which our factor of safety guide covers. Look at where the high values sit as well as their magnitude. A peak at the edge of a bolt hole or in a small fillet tells you where extra radius or material will do most good, and the stress concentration factors guide explains why those features attract stress.
Loads and supports should be applied the way the real part receives them: over a bolt head, a bearing face or a clamped area, rather than at a single point or along a perfectly rigid edge. Point loads and infinitely stiff supports produce local stresses the real part never sees.
When to bring in a specialist
Choosing the failure criterion is an engineering decision, shaped by the material, the way it is loaded and the code that applies. SixGate™’s stress analysis work reports stress in the form that suits your material and loading, against the code or standard you specify. High-stress regions come back as specific changes, such as a larger radius, a thicker web, a different alloy or a revised load path, with design variants compared side by side before you commit to one. The analysis is part of detail design and simulation, so the chosen variant moves on to its detailed model and manufacturing drawings, and into production, without changing hands.