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Von Mises equivalent stress from principal stresses

Von Mises equivalent stress from the three principal stresses.

Ref: MEC-65
Rev: v1
Field: Mechanical
Result: σ_VM [MPa]

σ_{VM} = \sqrt{\frac{{(σ_{1} - σ_{2})}^{2} + {(σ_{2} - σ_{3})}^{2} + {(σ_{3} - σ_{1})}^{2}}{2}}

σ_VM = √{[(σ₁ − σ₂)² + (σ₂ − σ₃)² + (σ₃ − σ₁)²]/2} depends only on the differences between the principal stresses, so a uniform all-round pressure, which changes the volume but not the shape, has no effect. Yield begins when σ_VM reaches the uniaxial yield strength S_y. For fixed principal stresses, σ_VM ≤ σ_T ≤ (2/√3) σ_VM, so von Mises gives an equal or higher factor of safety than Tresca. Include the zero out-of-plane stress when the state is plane stress.

Von Mises equivalent stress from the three principal stresses. It measures the shape-changing part of the stress state, so it ignores a uniform all-round pressure completely.

Calculator

Solve for

σ_VMVon Mises equivalent stressComputed

MPa

σ₁Principal stress 1

MPa

−200300

σ₂Principal stress 2

MPa

−200300

σ₃Principal stress 3

MPa

−200300

Tresca for the same principal stresses100 MPa
Ratio Tresca to von Mises1.125
Factor of safety for S_y = 250 MPa2.813

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

σ₁Principal stress 1Key driver

26316

σ₃Principal stress 3

60.6272

σ₂Principal stress 2

86.7293

The plane stress principal stresses 70, 0 and −30 MPa give σ_VM = 88.882 MPa and a factor of safety of 2.813 against 250 MPa, compared with 2.500 for Tresca. The three-dimensional example (110, 50, 20) gives 79.373 MPa.
Adding 100 MPa to every principal stress gives 88.882 MPa, unchanged. A hydrostatic state (H, H, H) gives 0 MPa, and uniaxial tension (S, 0, 0) gives S: 100 MPa for S = 100.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (MPa)	Local step	Change in result
σ₁ Principal stress 1	−200 to 300 MPa	26.03	316.1	290	+10 MPa	+10.8 %
σ₃ Principal stress 3	−200 to 300 MPa	60.64	271.8	211.2	+10 MPa	−7.91 %
σ₂ Principal stress 2	−200 to 300 MPa	86.67	293.1	206.4	+10 MPa	−1.92 %

Sweep of Principal stress 3

Principal stress 3 (MPa)	Von Mises equivalent stress (MPa)
−200	242.7
−137.5	182.8
−75	125.6
−12.5	77.01
50	62.45
112.5	98.39
175	152.6
237.5	211.4
300	271.8

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.

σ₁

Principal stress 1

MPa · typical −200 to 300

The first principal stress, in MPa. Tension is positive and compression negative; the slider runs from −200 to +300 MPa. Raises σ_VM when σ₁ moves further from the other two principal stresses. Raising it by 10 MPa (to 80 MPa) moves the result by +10.8 %, from 88.88 to 98.49 MPa.

σ₂

Principal stress 2

MPa · typical −200 to 300

The second principal stress, in MPa. In plane stress this is often the zero out-of-plane stress. The slider runs from −200 to +300 MPa. Raises σ_VM when σ₂ moves away from the midpoint of σ₁ and σ₃, and lowers it when it moves towards that midpoint. Tresca ignores the middle stress; von Mises does not. Raising it by 10 MPa (to 10 MPa) moves the result by −1.92 %, from 88.88 to 87.18 MPa.

σ₃

Principal stress 3

MPa · typical −200 to 300

The third principal stress, in MPa. Compression is negative. The slider runs from −200 to +300 MPa. Lowering it (more compressive) raises σ_VM, because it widens two of the three differences. Raising it by 10 MPa (to −20 MPa) moves the result by −7.91 %, from 88.88 to 81.85 MPa.

Worked example with the values above

Squared differences: (σ₁ − σ₂)² = 4,900, (σ₂ − σ₃)² = 900, (σ₃ − σ₁)² = 10,000.
Half the sum: (4,900 + 900 + 10,000)/2 = 7,900.
σ_VM = √7,900 = 88.882 MPa.

Assumptions and limits

The three values are the principal stresses of one state, tension positive.
An isotropic ductile material with equal tensile and compressive yield strength, under static load.

Common mistakes

Leaving out the zero out-of-plane stress in plane stress.
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 equivalent stress to cover load and material scatter. The sum of squares means errors in the largest difference dominate: a 3 MPa error in σ₁ shifts σ_VM by about 3 × (2σ₁ − σ₂ − σ₃)/(2σ_VM) = 2.9 MPa at the lecture values.

Where it is used

Distortion-energy yield check on principal stresses from any source.
Cross-checking a finite-element von Mises result.

Maths you need

Principal stresses: ordered or in any order
Distortion energy: the shape-changing part of the strain energy

Related formulas

Practise this formula