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Tresca equivalent stress from principal stresses
Tresca equivalent stress from the three principal stresses: the largest difference between any two of them, which is twice the largest shear stress.
- Ref
- MEC-64
- Rev
- v1
- Field
- Mechanical
- Result
- σT [MPa]
σT = max(|σ1 − σ2|, |σ2 − σ3|, |σ3 − σ1|) is the largest difference between any two principal stresses, so the order you enter them in does not matter. When they are sorted it reduces to σ1 − σ3, and the largest shear stress is τmax = σT/2. Yield begins when σT = Sy, equivalently τmax = Sy/2. Include the zero out-of-plane stress when the state is plane stress. The criterion is for pressure-insensitive ductile materials with equal tensile and compressive yield strength.
Tresca equivalent stress from the three principal stresses: the largest difference between any two of them, which is twice the largest shear stress. Yield begins when it reaches the uniaxial yield strength.
Calculator
- Maximum shear stress σT/250 MPa
- Von Mises for the same principal stresses88.88 MPa
- Factor of safety for Sy = 250 MPa2.5
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.
- The lecture plane stress state has principal stresses 70, 0 and −30 MPa (the zero included), so σT = 100 MPa and a factor of safety of 2.5 against a 250 MPa yield strength. The three-dimensional example (110, 50, 20) gives 90 MPa and 2.778.
- Adding the same pressure to all three principal stresses does not change σT: adding 100 MPa gives 100 MPa. Boundary checks: uniaxial tension S gives S, pure shear T gives 2T, hydrostatic stress gives 0.
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 | 30 | 330 | 300 | +10 MPa | +10 % |
| σ2 Principal stress 2 | −200 to 300 MPa | 100 | 330 | 230 | +10 MPa | 0 % |
| σ3 Principal stress 3 | −200 to 300 MPa | 70 | 300 | 230 | +10 MPa | −10 % |
Sweep of Principal stress 3
| Principal stress 3 (MPa) | Tresca equivalent stress (MPa) |
|---|---|
| −200 | 270 |
| −137.5 | 207.5 |
| −75 | 145 |
| −12.5 | 82.5 |
| 50 | 70 |
| 112.5 | 112.5 |
| 175 | 175 |
| 237.5 | 237.5 |
| 300 | 300 |
Sweep one variable
Variables and how each one contributes
Tresca equivalent stress
What the equation gives you. Everything else feeds it.
Principal stress 1
The first principal stress, in MPa. Tension is positive and compression negative; the slider runs from −200 to +300 MPa. Raises σT while σ1 is the largest value; it has no effect while σ1 lies between the other two. Raising it by 10 MPa (to 80 MPa) moves the result by +10 %, from 100 to 110 MPa.
Principal stress 2
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. Has no effect while σ2 lies between σ1 and σ3, because Tresca depends only on the largest and smallest. Raising it by 10 MPa (to 10 MPa) moves the result by +0 %, from 100 to 100 MPa.
Principal stress 3
The third principal stress, in MPa. Compression is negative. The slider runs from −200 to +300 MPa. Lowering it (more compressive) raises σT while σ3 is the smallest value. Raising it by 10 MPa (to −20 MPa) moves the result by −10 %, from 100 to 90 MPa.
Worked example with the values above
- The three differences: |σ1 − σ2| = 70, |σ2 − σ3| = 30, |σ3 − σ1| = 100 MPa.
- Take the largest.
- σT = 100 MPa, so τmax = 50 MPa.
Assumptions and limits
- The three values are the principal stresses of one stress 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 when two in-plane principal stresses have the same sign.
- Using |σ1| − |σ3| instead of the difference with signs.
Statistics and uncertainty
σT is a difference of two principal stresses, so independent errors of ±3 MPa in each give about ±4.2 MPa in σT, roughly 4.2 % of the lecture value. The middle stress contributes no error while it stays between the other two.
Where it is used
- Maximum-shear yield check on principal stresses from hand calculation, a finite-element program or measurement.
- Comparing with von Mises for the same principal stresses.
Maths you need
- Principal stresses: the stresses on planes with no shear
- Absolute value: size without sign