Von Mises stress calculator
Von Mises stress, also called equivalent stress, combines all the stress components at a point into one value that can be compared with the yield strength of an isotropic, ductile material. Enter the stresses as a full 3D tensor, in plane stress or as principal stresses, and get the von Mises stress, the principal stresses and the safety factor against yield.
For a hand check of one stress state. The results assume an isotropic, ductile material under static load and are only as good as the stresses you enter. They do not replace a full analysis or the checks your design code requires, and you remain responsible for the inputs and any design decision based on them. The unit buttons label the values; they do not convert them.
1Stresses at the point
Tension is positive, compression negative. Use the same unit for every value. Empty fields count as zero. Type numbers without thousands separators (10000, not 10,000).
Von Mises stress σvM
- Principal stresses σ1 · σ2 · σ3
- –
- Tresca equivalent stress σ1 − σ3
- –
- Maximum shear stress (σ1 − σ3)/2
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- Mean (hydrostatic) stress (σ1 + σ2 + σ3)/3
- –
- Signed von Mises (sign of σ1)
- –
σ1 is zero here, so the sign is taken from the compressive principal stresses.
2Yield check (optional)
The minimum specified yield strength of your material (ReH, or the 0.2 % proof stress Rp0.2 if there is no distinct yield point), at the operating temperature and for the product thickness, in the same unit as the stresses.
The von Mises and Tresca criteria are for ductile materials that fail by yielding. Brittle materials such as grey cast iron or ceramics need a brittle failure theory instead, such as the maximum normal stress or the Coulomb–Mohr theory (see below).
Safety factor against yield n = Sy/σvM
- Safety factor with Tresca Sy/(σ1 − σ3)
- –
- Utilisation σvM/Sy, in %
- –
Enter a yield strength to see the safety factor.
3Where the stress state sits on the yield surface
The plot needs a yield strength; every stress is divided by Sy. Inside a curve, that criterion predicts elastic behaviour; on or outside it, yield. Between the Tresca hexagon and the von Mises curve, Tresca predicts yield and von Mises does not.
In plane stress the axes are the two in-plane principal stresses σA and σB. For a 3D state without a zero principal stress the plot shows the deviatoric plane instead, the view along the hydrostatic axis, where von Mises is a circle and Tresca a hexagon.
How to use the calculator
Choose how you know the stress state: all six components of the stress tensor, plane stress (σx, σy, τxy) or the three principal stresses. Use one unit for every value; tension is positive, compression negative. In a beam, σx is the bending stress and τxy the shear stress, and for a standard section our beam calculator works both out from the bending moment and the shear force.
Step 1 gives the von Mises stress, the principal stresses sorted from largest to smallest, the Tresca equivalent stress, the maximum shear stress, the mean (hydrostatic) stress and a signed von Mises stress.
Add the minimum specified yield strength of your material in step 2 to see the safety factor against yield, by von Mises and by Tresca. Step 3 plots the stress state against both yield surfaces: in the plane of the other two principal stresses when one principal stress is zero, as in plane stress, otherwise in the deviatoric plane.
“Copy the results” puts the inputs and results on your clipboard as plain numbers, for a report or a spreadsheet. The calculator runs in your browser and does not send the values you enter anywhere.
Limitations. The calculator evaluates one stress state with the von Mises and Tresca yield criteria, for an isotropic, ductile material under static load. It does not check fatigue, buckling, fracture or stresses at singular points of a finite element model, and it does not apply the partial factors or allowable stresses of a design code. Enter the local stress at the point, including any stress concentration. Use the results to check a hand calculation or a value read from an FE model, not as the only basis for a design decision. Volupe provides the calculator free of charge and without warranty.
What is von Mises stress?
Von Mises stress, also called equivalent or effective stress, is a single number calculated from all the stress components at a point. It measures the part of the stress that changes the shape of the material rather than its volume, which is what makes ductile metals yield.
In a tensile test the von Mises stress equals the applied stress, so it can be compared directly with the yield strength Sy. A ductile material is predicted to yield when its von Mises stress reaches Sy.
Equal pressure in every direction does not change the von Mises stress: a purely hydrostatic stress state gives zero. Because the formula squares the stress differences, the von Mises stress is never negative and cannot tell tension from compression.
The von Mises yield criterion and the safety factor
When a ductile material yields
The von Mises criterion predicts yield when the von Mises stress reaches the yield strength from a simple tensile test. It is the usual yield measure for many ductile engineering materials, such as structural metals, and most commercial finite element programs compute it as a standard result.
It is also called the distortion energy or octahedral shear stress criterion: the von Mises stress is the uniaxial stress that would store the same distortion energy as the actual combination of stresses, and it is proportional to the shear stress on the plane equally inclined to the three principal axes.
Why pressure alone does not cause yielding
Adding the same pressure in every direction leaves the von Mises stress unchanged, and tests show that hydrostatic stress has practically no effect on yielding in metals. That does not make hydrostatic tension harmless: yield is driven by shear stresses and fracture by normal stresses, so a state with large tensile stresses in all three directions can lead to fracture with little or no yielding. That needs a fracture assessment, not a yield check.
Polymers behave differently: they resist yielding better in compression than in tension, so plain von Mises suits them less well.
Brittle materials need a different check
Ductile materials fail by yielding, which is controlled by shear, so von Mises or Tresca applies. Brittle materials such as grey cast iron crack or crush and are usually limited by their tensile strength: use the maximum normal stress theory or the Coulomb–Mohr or modified Mohr theories instead.
Safety factor against yield
The safety factor against yield is the yield strength divided by the von Mises stress, n = Sy/σvM: the factor by which all the stresses would have to grow together to reach yield. A value above 1 means the calculated stress is below yield. Many design rules limit the calculated stress to an allowable value below yield; the margin required, and whether local yielding is acceptable, depend on the design code, the type of stress and the load case.
Von Mises or Tresca?
Two criteria for the same question
The Tresca (maximum shear stress) criterion predicts yield when the largest shear stress reaches half the yield strength. That is the same as the largest difference between two principal stresses, σ1 − σ3, reaching Sy.
Pure shear shows the largest difference
In pure shear, von Mises predicts yield at a shear stress of Sy/√3 ≈ 0.577 Sy and Tresca at 0.5 Sy. Put the other way round, the von Mises stress in pure shear is √3 times the shear stress.
The two criteria never differ by more than about 15 %. The difference is largest when the middle principal stress lies halfway between the other two, σ2 = (σ1 + σ3)/2, as in pure shear: there the Tresca stress is 2/√3 ≈ 1.155 times the von Mises stress. Test data scatter: von Mises often fits them somewhat better, but neither is a clear winner.
Which one to use
Tresca is slightly more conservative: it predicts yield earlier. Its yield surface has corners, which makes it less convenient in finite element codes, and finite element analysis (FEA) programs usually report von Mises. The calculator gives both.
In the calculator
Step 2 gives the safety factor by both criteria. Step 3 plots your stress state against both yield surfaces: in the plane of the other two principal stresses when one principal stress is zero, as in plane stress, otherwise in the deviatoric plane, the view along the hydrostatic axis, where von Mises is a circle and Tresca a hexagon.
Von Mises stress formulas
The formulas behind the results. σ1 ≥ σ2 ≥ σ3 are the principal stresses, sorted from largest to smallest; σA and σB are the two in-plane principal stresses of a plane stress state, in any order.
σvM = √(½[(σx − σy)² + (σy − σz)² + (σz − σx)² + 6(τxy² + τyz² + τzx²)]) (1)
From all six components of the stress tensor.
σvM = √(½[(σ1 − σ2)² + (σ2 − σ3)² + (σ3 − σ1)²]) (2)
From principal stresses, in any order.
σvM = √(σx² − σxσy + σy² + 3τxy²) = √(σA² − σAσB + σB²) (3)
Plane stress: σz = τyz = τzx = 0. σA, σB = (σx + σy)/2 ± √(((σx − σy)/2)² + τxy²) are the two in-plane principal stresses, in any order; the third principal stress is zero.
J2 = (1/6)[(σ1 − σ2)² + (σ2 − σ3)² + (σ3 − σ1)²], σvM = √(3J2) (4)
J2 is the second invariant of the deviatoric stress, the stress minus its mean (hydrostatic) part.
λ³ − I1λ² + I2λ − I3 = 0 (5)
The principal stresses are its three roots, with I1 = σx + σy + σz, I2 = σxσy + σyσz + σzσx − τxy² − τyz² − τzx² and I3 the determinant of the stress matrix.
σT = σ1 − σ3 = max(|σ1 − σ2|, |σ2 − σ3|, |σ3 − σ1|) (6)
Tresca equivalent stress; the maximum shear stress is half of it. In plane stress the zero out-of-plane stress counts: σT = max(|σA|, |σB|, |σA − σB|), so for σA = 200 and σB = 100 it is 200, not 100.
n = Sy/σvM, nT = Sy/(σ1 − σ3) (7)
Safety factors against yield by von Mises and by Tresca.
How the calculator solves it: the von Mises stress comes straight from the six components (formula (1), which equals √(3J2)). The principal stresses are the eigenvalues of the symmetric stress matrix, found with Jacobi rotations, an iterative method. Plane stress assumes σz = τyz = τzx = 0. The tests under “How the calculator was checked” confirm that both routes agree.
Worked example: a rod in tension and torsion
Take a solid rod, 40 mm in diameter, that carries an axial tensile force of 150 kN and a torque of 1.0 kN·m, and assume a yield strength of 355 MPa. The most stressed points are on the outer surface, where the stress state is plane stress: the axial force gives σx = P/A with A = πd²/4, and the torque gives τxy = 16T/(πd³) at the surface.
| Quantity | Value | How |
|---|---|---|
| Normal stress from the axial force, σx | 119.4 MPa | from P/A, A = πd²/4 |
| Shear stress from the torque, τxy | 79.6 MPa | from 16T/(πd³) |
| In-plane principal stresses σA, σB | 159.2 and −39.8 MPa | from the σA, σB formula under (3) |
| Von Mises stress σvM | 182.4 MPa | from formula (3) |
| Tresca equivalent stress σT | 199.0 MPa | from max(|σA|, |σB|, |σA − σB|) |
| Safety factor, von Mises | 1.95 | 355/182.4 |
| Safety factor, Tresca | 1.78 | 355/199.0 |
Values from the calculator, with the stresses rounded to 0.1 MPa before they are entered. With the unrounded stresses, 119.37 and 79.58 MPa, the von Mises stress is 182.3 MPa and the Tresca stress 198.9 MPa; the safety factors are the same.
Both criteria predict no yield, and Tresca gives the lower safety factor, as expected for the more conservative criterion. “Fill with example values” at the top loads this case into the calculator, in plane stress.
Reading von Mises stress in FEA results
It has no sign
A von Mises plot cannot show whether a region is in tension or compression. Siemens describes a signed von Mises result in Simcenter 3D: the von Mises value multiplied by the sign of the maximum principal stress, built with result variables and a result probe. The sign follows the largest principal stress alone, so a mostly compressive state with a small tensile σ1 still shows as positive: check the principal stresses before drawing conclusions. The calculator shows this signed value too. When σ1 is zero, as in uniaxial compression or a plate compressed in both directions, the calculator shows the value as negative. That is the calculator’s own convention for that case; the Siemens formula always takes the sign of the maximum principal stress.
Peaks at sharp corners and point loads
At a crack tip the exact elastic stress is unbounded: a stress singularity. Sharp inside (re-entrant) corners give singular stresses too, and point loads create stress singularities in a finite element model. Ordinary finite elements cannot reproduce a singularity, so a peak von Mises value at such a point is not a design value.
Averaged or unaveraged stresses
In a standard finite element model the stresses jump from one element to the next, so contour plots show post-processed values; the simplest method averages neighbouring elements at each node. Averaging across a junction between parts, such as a stiffener and a panel, can give wrong values there, so compare averaged and element results before trusting a peak. In a Simcenter 3D result variable you choose where a result is taken (elements, element nodes or nodes) and whether it is averaged.
Fatigue can use other stress measures
For fatigue, the stress tensor can be reduced to a single value in other ways than the plain von Mises stress. Simcenter SimSolid, for example, offers signed von Mises, maximum principal, absolute maximum principal, signed maximum shear and critical-plane methods for uniaxial fatigue, and uses the full stress tensor for multiaxial fatigue.
Safety factor and yield in Simcenter 3D and Simcenter Femap
In Simcenter 3D a result probe can compute the safety factor of every element as the yield strength divided by the element von Mises stress: the same n = Sy/σvM as this calculator. Viktor Hultgren shows how in Customize results with result probes in Simcenter 3D. In Simcenter Femap, von Mises can be set as the yield criterion of an elastic-plastic material in a nonlinear analysis, as in a Siemens post-buckling example.
From one point to the whole part
This calculator checks one stress state. A finite element model computes it everywhere in the part; our guide to finite element analysis explains how, and how to check the result. To learn the workflow in Simcenter 3D, see Simcenter 3D training.
Where the von Mises criterion comes from
In 1904 M. T. Huber, a Polish professor, proposed strain energy as the measure of how close a material is to failure. Because volume change under compression does not cause failure, he argued, the distortion energy alone is probably the right measure when the material is compressed.
In 1913 Richard von Mises proposed replacing the hexagon of the maximum shear stress condition with a simpler curve, the circle that circumscribes it. That smooth condition is the von Mises criterion; seen in plane stress, as in the figure above, the circle becomes the ellipse.
Huber published in Polish, so few people read his work at first. It became known through A. and L. Föppl’s book Drang und Zwang (1920) and a discussion between Huber, von Mises and Hencky at the First Congress of Mechanics in Delft in 1924, the year Heinrich Hencky published a paper on the theory of plastic deformation in ZAMM.
Today the criterion is often called the Huber–Mises–Hencky criterion, after all three, or simply von Mises.
References and changelog
Sources
D. Roylance, MIT, Yield and plastic flow (2001, PDF) and Shear and torsion (2000, PDF, the torsion formula of the worked example); T. Wierzbicki, MIT, 2.080J Structural mechanics, lecture 12: Fundamental concepts in structural plasticity (2013, PDF).
R. Radovitzky, MIT, 16.001 Unified engineering, lectures 11–12: Transformation of stress components, principal and maximum shear stresses, Mohr’s circle (2021, PDF); T. A. Jaeger, MIT, 22.314J lecture note M-5A: Some introductory notes to the theory of plasticity (2006, PDF).
P. Pinsky, Stanford, ME111 class 21: Failure of ductile materials under static loading (2000, PDF, used to cross-check the calculator); WPI, ME-3320 Design of machine elements, lectures 9–10: Static failure theories (2025, PDF).
NASA NTRS: A. Tessler, H. R. Riggs and M. Dambach, A novel four-node quadrilateral smoothing element for stress enhancement and error estimation (AIAA paper 98-1713, 1998, PDF); R. Plunkett, Convergence rates for finite element problems with singularities, part 1: Antiplane shear (NASA CR-164633, 1980, PDF); J. L. Swedlow, Singularity computations (Carnegie-Mellon University report SM-76-6, 1976, PDF). K. J. Bathe, MIT, Finite Element Procedures – En Plus (in press with Springer, 2025; PDF on the author’s MIT page).
Siemens: Tutorial: How to read Stress Signed (Simcenter 3D, Siemens Support Center, KB000129795, 2024); Post-buckling in Femap using Advanced Nonlinear (Simcenter blog, 2016). Altair: Fatigue assessment methods, Altair SimSolid User Guide 2026.
History: M. T. Huber, Specific work of strain as a measure of material effort (Czasopismo Techniczne, Lwów, 1904; English translation in Archives of Mechanics 56(3), 2004); M. Życzkowski, Preface, Archives of Mechanics 57(2–3), 2005.
R. v. Mises, The mechanics of solids in the plastically-deformable state (NASA TM-88448, 1986, translation of the 1913 paper); H. Hencky, Zur Theorie plastischer Deformationen und der hierdurch im Material hervorgerufenen Nachspannungen (ZAMM 4(4), 1924).
Simcenter 3D example: Volupe blog, Customize results with result probes in Simcenter 3D (2025). Illustrations drawn by Volupe; the yield-surface plots are drawn from the formulas.
How the calculator was checked
The calculator was checked against 27 test cases computed independently (uniaxial, pure shear, hydrostatic, plane stress, full 3D tensors, principal-stress input, very large and very small magnitudes, and the Stanford ME111 bracket example). It reproduces the Stanford values; the one that the handout prints as 5274 is 5276 by the formulas, a slip in the handout. In 200 000 random stress tensors, the principal stresses also reproduced the invariants and the von Mises stress. There were no failures.
Changelog
October 2026: first version. 26 October 2026: limitations, triaxial-tension and grey cast iron notes, plot description, rounding near a safety factor of 1, Tresca verdict at exactly yield.
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Questions about von Mises stress
Simcenter 3D
Viktor Hultgren
Application Specialist, M.Sc.
Former Altair tools
Johan Dahlberg
Application Support Specialist, Ph.D.
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Is von Mises stress the same as equivalent stress?
Usually. Von Mises stress is also called equivalent or effective stress: a single value that combines all the stress components at a point. In a tensile test it equals the applied stress, so it can be compared directly with the yield strength. The exception is the Tresca equivalent stress, σ1 − σ3, another equivalent stress, which the calculator gives as well.
How do you calculate von Mises stress?
From the principal stresses: σvM = √(½[(σ1 − σ2)² + (σ2 − σ3)² + (σ3 − σ1)²]). With all six components use formula (1) on this page; in plane stress, formula (3) with σA and σB, the two in-plane principal stresses. The calculator at the top does all three.
How is the safety factor against yield calculated?
Divide the yield strength by the von Mises stress: n = Sy/σvM. Above 1, the calculated stress is below yield. The margin required depends on the design code, the type of stress and the load case.
What is the difference between von Mises and Tresca?
Tresca predicts yield when the largest shear stress reaches half the yield strength; von Mises uses the distortion energy. Tresca is slightly more conservative, and the two never differ by more than about 15 %, most in pure shear.
Can von Mises stress be negative?
No. It treats tension and compression alike and is never negative. In Simcenter 3D you can build a signed von Mises result with result variables and a result probe: the von Mises value with the sign of the maximum principal stress.
Should I check von Mises or maximum principal stress?
Von Mises for ductile materials, which fail by yielding. Brittle materials such as grey cast iron crack rather than yield and are usually limited by their tensile strength, so use a brittle failure theory instead: the maximum normal (maximum principal) stress theory, or the Coulomb–Mohr or modified Mohr theory. The calculator gives the principal stresses as well.
Why is the von Mises stress so high at a sharp corner in my FE model?
A sharp inside (re-entrant) corner, a crack tip or a point load creates a stress singularity: the exact elastic stress at that point is unbounded and ordinary finite elements cannot reproduce it, so the peak value is not a design value. Our FEA guide explains how to check results.
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