BIZOMAT 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. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: σx
- What it is
- The direct (normal) stress acting on the face square to the X axis at the point, from hand calculation, strain gauges or finite element results.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts -100000 MPa to 100000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 80 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Tension is positive and compression negative; keep the same sign rule for every input.
What this means: σy
- What it is
- The direct (normal) stress acting on the face square to the Y axis at the same point.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts -100000 MPa to 100000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with -30 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Tension is positive and compression negative.
What this means: σz
- What it is
- The direct (normal) stress acting on the face square to the Z axis at the same point. Use 0 for a plane-stress state.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts -100000 MPa to 100000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Tension is positive and compression negative.
What this means: τxy
- What it is
- The shear stress acting in the XY plane: on the X face in the Y direction, equal to that on the Y face in the X direction.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts -100000 MPa to 100000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 40 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Shear signs must follow one consistent tensor convention, as in your analysis software.
What this means: τyz
- What it is
- The shear stress acting in the YZ plane: on the Y face in the Z direction, equal to that on the Z face in the Y direction.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts -100000 MPa to 100000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Shear signs must follow one consistent tensor convention.
What this means: τzx
- What it is
- The shear stress acting in the ZX plane: on the Z face in the X direction, equal to that on the X face in the Z direction.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts -100000 MPa to 100000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with -15 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Shear signs must follow one consistent tensor convention.
What this means: Yield strength
- What it is
- The minimum yield strength of the material, used to give the safety factor against yield from the von Mises stress.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 100000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 350 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The safety factor is directly proportional to the yield strength; the stresses do not change.
- Get it right
- Use the guaranteed minimum from the certificate or specification, not a typical value.
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.
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Method explained here · BIZOMAT original book
Three-dimensional stress
In Stress Transformation by the Numbers
What it covers: How to find the three principal stresses of a general stress state from its invariants, how three Mohr's circles show the true maximum shear, and when ignoring a direction misleads.
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