Library / Mechanical / MEC-56

Mechanical

Tresca to von Mises ratio

How much more severe the Tresca equivalent stress is than the von Mises equivalent stress for the same state of stress.

Ref: MEC-56
Rev: v1
Field: Mechanical
Result: σ_T/σ_VM

\frac{σ_{T}}{σ_{VM}} = \frac{σ_{1} - σ_{3}}{\sqrt{{σ_{x}}^{2} - σ_{x} σ_{y} + {σ_{y}}^{2} + 3 {τ_{xy}}^{2}}}

For the same stress state Tresca is never lower than von Mises, and never more than 2/√3 ≈ 1.1547 times as high: σ_VM ≤ σ_T ≤ (2/√3) σ_VM. They agree for uniaxial tension and for equal biaxial tension (ratio 1), and differ most for pure shear (ratio 1.1547). Tresca therefore gives an equal or lower factor of safety. The ratio is the quickest way to see how much extra conservatism Tresca carries for a given state; in pure shear the Tresca allowable shear is 13.4 % lower than von Mises (or von Mises is 15.5 % higher than Tresca): state which one you divide by when quoting percentages.

How much more severe the Tresca equivalent stress is than the von Mises equivalent stress for the same state of stress. It lies between 1 and 2/√3, about 1.155, for every stress state.

Calculator

Solve for

σ_T/σ_VMTresca to von Mises ratioComputed

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

Tresca equivalent stress σ_T100 MPa
Von Mises equivalent stress σ_VM88.88 MPa
Upper limit of the ratio, 2/√31.155

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.

Tresca to von Mises ratio () current value 1.125

σ_xNormal stress in x (tension positive)Key driver

11.15

σ_yNormal stress in y (tension positive)

11.15

τ_xyShear stress on the x face in the y direction

1.081.15

The lecture state gives a ratio of 1.125: Tresca is 12.5 % more severe here. In uniaxial tension the ratio is 1 and in pure shear it is 1.155, the largest possible.
Equal biaxial tension of 100 MPa also gives 1 because the zero out-of-plane stress is included in the Tresca difference.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing ()	Local step	Change in result
σ_x Normal stress in x (tension positive)	−200 to 300 MPa	1.001	1.153	0.1524	+10 MPa	−0.955 %
σ_y Normal stress in y (tension positive)	−200 to 300 MPa	1.005	1.155	0.1498	+10 MPa	−1.86 %
τ_xy Shear stress on the x face in the y direction	−150 to 150 MPa	1.078	1.151	0.07379	+6 MPa	+0.436 %

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

Shear stress on the x face in the y direction (MPa)	Tresca to von Mises ratio
−150	1.151
−112.5	1.149
−75	1.143
−37.5	1.123
0	1.078
37.5	1.123
75	1.143
112.5	1.149
150	1.151

Sweep one variable

Vary

Tresca to von Mises ratio ()Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

σ_T/σ_VM

Tresca to von Mises ratio

Result ·

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. Moves the state towards uniaxial tension or towards pure shear. The ratio is 1 for the first and 1.155 for the second; here a rise in σ_x moves it towards tension. Raising it by 10 MPa (to 60 MPa) moves the result by −0.955 %, from 1.125 to 1.114.

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. Moves the state towards uniaxial tension or towards pure shear. The ratio is 1 for the first and 1.155 for the second; here a rise from −10 towards zero moves it towards uniaxial tension. Raising it by 10 MPa (to 0 MPa) moves the result by −1.86 %, from 1.125 to 1.104.

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. Pushes the ratio towards the pure-shear value, 1.155. The sign of the shear does not matter. Raising it by 10 MPa (to 50 MPa) moves the result by +0.677 %, from 1.125 to 1.133.

Worked example with the values above

Tresca: σ_T = σ₁ − σ₃ = 70 − (−30) = 100 MPa.
Von Mises: σ_VM = 88.882 MPa.
Ratio: 100 / 88.882 = 1.1251.

Assumptions and limits

Plane stress, so the zero out-of-plane stress is a principal stress.
An isotropic ductile material with equal tensile and compressive yield strength.

Common mistakes

Quoting a percentage difference without saying which equivalent stress is the denominator.
Expecting the ratio to exceed 1.1547, or to fall below 1.

Statistics and uncertainty

The 15 % maximum difference is usually smaller than the scatter in measured yield strength, so the choice between criteria is secondary to the safety factor chosen. Experiments on ductile metals fall between the two criteria, nearer to von Mises for most steels and aluminium.

Where it is used

Deciding whether the extra conservatism of Tresca matters for a given load case.
Cross-checking a hand calculation against a finite-element von Mises result.

Maths you need

Tresca: σ₁ − σ₃
Von Mises: distortion-energy equivalent stress

Related formulas

Practise this formula