KEVOS® MAKEOS · Strength
Three-Dimensional Principal Stresses
What are the three principal stresses, maximum shear and von Mises stress for this 3D stress state?
Principal stresses of a general three-dimensional stress state from its six components, by the stress invariants, with maximum shear, von Mises equivalent stress and the safety 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. |
| Yield strength | MPa | Minimum yield strength from the material certificate or specification. |
Method and formulas
- Invariants I₁ = σx + σy + σz; I₂ = σxσy + σyσz + σzσx − τxy² − τyz² − τzx²; I₃ = det of the stress tensor.
- Principal stresses are the roots of σ³ − I₁σ² + I₂σ − I₃ = 0 (trigonometric solution).
- τmax = (σ₁ − σ₃)/2; von Mises = √(½[(σ₁ − σ₂)² + (σ₂ − σ₃)² + (σ₃ − σ₁)²]).
- Example: the default state has principal stresses 94.711, 21.367 and -46.078 MPa.
Limitations
- Static, isotropic material for the yield criteria.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Post-process FEA or strain gauge results
- Check triaxial stress points in thick parts
- Teach and check stress transformation
Who it is for
- Design engineers
- Structural and mechanical engineers
- Engineering students
What you get
- Principal stresses σ1 ≥ σ2 ≥ σ3
- Maximum shear stress
- von Mises stress and safety factor
Interpreting the result
- Hydrostatic stress (equal principal stresses) does not cause yield in ductile metals.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 212–218. Topic reference; no source tables or recommended settings are reproduced.
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