KEVOS® MAKEOS · Strength
Triaxial Stress Check (Principal Stresses, von Mises, Tresca, Strains)
For this three-dimensional stress state, what are the principal stresses, how close is it to yield by von Mises and by Tresca, and what strains will a gauge or a fit see?
Check of a general three-dimensional stress state: principal stresses from the stress invariants, maximum shear and von Mises stress (as the three-dimensional principal stress tool), the safety factor against yield by von Mises and by Tresca, shear and bulk moduli from E and Poisson’s ratio (as the elastic constants tool), the principal strains by Hooke’s law and the volume change, against your required factor.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| σx | MPa | Normal stress in x (tension positive). |
| σy | MPa | Normal stress in y. |
| σz | MPa | Normal stress in z. |
| τxy | MPa | Shear stress in the xy plane. |
| τyz | MPa | Shear stress in the yz plane. |
| τzx | MPa | Shear stress in the zx plane. |
| Material | ||
| Elastic modulus | GPa | Elastic modulus. |
| Poisson’s ratio | Poisson’s ratio. | |
| Yield strength | MPa | Yield strength in tension. |
| Required safety factor | Safety factor you need against yield. | |
Method and formulas
- Principal stresses are the roots of σ³ − I₁σ² + I₂σ − I₃ = 0; von Mises = √{½[(σ₁ − σ₂)² + (σ₂ − σ₃)² + (σ₃ − σ₁)²]} (as the three-dimensional principal stress tool).
- Factor by von Mises = S_y ÷ σ_vm; by Tresca = S_y ÷ (σ₁ − σ₃).
- G = E ÷ 2(1 + ν); K = E ÷ 3(1 − 2ν) (as the elastic constants tool); ε₁ = [σ₁ − ν(σ₂ + σ₃)] ÷ E and so on; volume change = (1 − 2ν) I₁ ÷ E.
- Example: principal stresses 94.711, 21.367 and -46.078 MPa; von Mises 121.96 MPa gives a factor of 2.8697 (Tresca 2.486).
Limitations
- Isotropic, linear-elastic material.
- Static stress; cyclic states need a fatigue check.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check a point in a thick part, fit or pressure component
- Compare von Mises and Tresca margins
- Convert a stress state to strains for a gauge check
Who it is for
- Stress engineers
- Design engineers
- Students
What you get
- Principal stresses, maximum shear and von Mises stress
- Safety factors by von Mises and Tresca
- Principal strains, volume change, G and K
Interpreting the result
- Tresca is always the more conservative of the two; the difference is largest in pure shear.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 212–218, 369–380. Topic reference; no source tables or recommended settings are reproduced.
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Open the Triaxial Stress Check (Principal Stresses, von Mises, Tresca, Strains) calculator
