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Mechanical

Von Mises equivalent stress

Single stress value used to compare a combined stress state with the yield strength.

Ref: MEC-25
Rev: v1
Field: Mechanical
Result: σ_VM [MPa]

σ_{VM} = \sqrt{{σ_{x}}^{2} - σ_{x} σ_{y} + {σ_{y}}^{2} + 3 {τ_{xy}}^{2}}

Von Mises combines all the stress components into one equivalent tensile stress, based on the distortion energy: yielding begins when σ_VM reaches the uniaxial yield strength S_y. For plane stress it can be written straight from the components, σ_VM = √(σ_x² − σ_xσ_y + σ_y² + 3τ_xy²), or from the sorted principal stresses (the zero included), which gives exactly the same value. A uniform all-round pressure has no effect on it, because it changes volume, not shape. For pure shear it reduces to √3|τ_xy|. Step 5 of the stress route, alongside Tresca.

Single stress value used to compare a combined stress state with the yield strength. It is built from the distortion energy, so a uniform all-round pressure has no effect on it.

Calculator

Solve for

σ_VMVon Mises equivalent stressComputed

MPa

σ_xNormal stress in x (tension positive)

MPa

−200300

σ_yNormal stress in y (tension positive)

MPa

−200300

τ_xyShear stress on the x face in the y direction

MPa

−150150

Ratio to a 250 MPa yield strength (utilisation at n = 1)0.3555
From the principal stresses (should be the same)88.88 MPa
Pure-shear equivalent √3 τ69.28 MPa
Tresca for comparison100 MPa

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Von Mises equivalent stress (MPa) current value 88.88 MPa

σ_xNormal stress in x (tension positive)Key driver

70313

τ_xyShear stress on the x face in the y direction

55.7266

σ_yNormal stress in y (tension positive)

82.1287

The lecture state gives 88.882 MPa, the same value as √[((σ₁ − σ₂)² + (σ₂ − σ₃)² + (σ₃ − σ₁)²)/2] with σ_1,2,3 = 70, 0, −30 MPa. Against a 250 MPa yield strength the factor of safety is 2.813.
Equal biaxial tension of 100 MPa gives σ_VM = 100 MPa, no more than the tension in either direction, and equal biaxial tension and compression (100, −100) gives 173.2 MPa, a much more severe state.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (MPa)	Local step	Change in result
σ_x Normal stress in x (tension positive)	−200 to 300 MPa	70	312.9	242.9	+10 MPa	+7.33 %
τ_xy Shear stress on the x face in the y direction	−150 to 150 MPa	55.68	265.7	210	+6 MPa	+9.36 %
σ_y Normal stress in y (tension positive)	−200 to 300 MPa	82.12	286.9	204.8	+10 MPa	−3.87 %

Sweep of Shear stress on the x face in the y direction

Shear stress on the x face in the y direction (MPa)	Von Mises equivalent stress (MPa)
−150	265.7
−112.5	202.7
−75	141.3
−37.5	85.55
0	55.68
37.5	85.55
75	141.3
112.5	202.7
150	265.7

Sweep one variable

Vary

Von Mises equivalent stress (MPa)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

σ_VM

Von Mises equivalent stress

Result · MPa

What the equation gives you. Everything else feeds it.

σ_x

Normal stress in x (tension positive)

MPa · typical −200 to 300

Normal stress on the x face, in MPa. Tension is positive and compression negative, so the slider runs from −200 to +300 MPa. Raises σ_VM at the default state. The slope is (2σ_x − σ_y)/(2σ_VM), so it turns negative only if σ_x falls below σ_y/2. Raising it by 10 MPa (to 60 MPa) moves the result by +7.33 %, from 88.88 to 95.39 MPa.

σ_y

Normal stress in y (tension positive)

MPa · typical −200 to 300

Normal stress on the y face, in MPa. Tension is positive and compression negative, so the slider runs from −200 to +300 MPa. Lowers σ_VM at the default state, where σ_y is negative and opposes σ_x. The slope is (2σ_y − σ_x)/(2σ_VM), so it turns positive once σ_y exceeds σ_x/2. Raising it by 10 MPa (to 0 MPa) moves the result by −3.87 %, from 88.88 to 85.44 MPa.

τ_xy

Shear stress on the x face in the y direction

MPa · typical −150 to 150

Shear stress on the x face acting in the +y direction, in MPa. The sign matters for directions and rotated stresses, so the slider runs from −150 to +150 MPa. Raises σ_VM with weight 3 inside the root. The sign of the shear does not matter, since only its square appears. Raising it by 10 MPa (to 50 MPa) moves the result by +15.8 %, from 88.88 to 103 MPa.

Worked example with the values above

Normal-stress part: σ_x² − σ_xσ_y + σ_y² = 3,100. Shear part: 3τ_xy² = 3 × 40² = 4,800.
Sum: 7,900.
σ_VM = √7,900 = 88.88 MPa.

Assumptions and limits

Plane stress, an isotropic ductile material with equal tensile and compressive yield strength.
Static loading; fatigue, brittle fracture and buckling are separate checks.

Common mistakes

Using the maximum shear stress without the factor 3 on the shear term.
Comparing with the ultimate rather than the yield strength.

Statistics and uncertainty

Safety factors of 1.5 or more are normally applied on top of the calculated stress to cover load and material scatter. The shear term is weighted by 3, so a 5 % error in τ_xy changes σ_VM by a different amount from the same error in a normal stress, depending on the state.

Where it is used

Yield checks in structures and pressure vessels.
Interpreting finite-element results, which usually report von Mises stress.

Maths you need

Principal stress: plane stress and the zero out-of-plane value
Yield strength: onset of permanent deformation

Related formulas

Practise this formula