BIZOMAT 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. | |
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 normal stress acting on the face square to the x axis, at the point being checked, from your stress analysis.
- 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, compression negative; keep the signs consistent for all six components.
What this means: σy
- What it is
- The 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 positive, compression negative.
What this means: σz
- What it is
- The normal stress acting on the face square to the z 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 20 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter zero for a plane stress state, such as a free surface.
What this means: τxy
- What it is
- The shear stress acting in the xy plane at the point, on the x face in the y direction (and equally 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
- Use the sign convention of your analysis; a wrong sign changes the principal stresses.
What this means: τyz
- What it is
- The shear stress acting in the yz plane at the point, on the y 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 10 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the same sign convention as the other components.
What this means: τzx
- What it is
- The shear stress acting in the zx plane at the point, on the z 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 -15 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the same sign convention as the other components.
Material
What this means: Elastic modulus
- What it is
- The elastic (Young’s) modulus of the material, used only for the strains and the shear and bulk moduli; the stresses do not depend on it.
- Units and format
- Enter a number in GPa: gigapascals (1 GPa = 1000 MPa). The form accepts 0.000001 GPa to 2000 GPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 200 GPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter gigapascals, not megapascals.
What this means: Poisson’s ratio
- What it is
- Poisson’s ratio of the material: the ratio of lateral to axial strain, from the material data sheet. It is used for the strains and moduli.
- Units and format
- Enter a plain number with no unit. The form accepts 0 to 0.4999. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.3 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- A dimensionless ratio below one half, not a percentage.
What this means: Yield strength
- What it is
- The yield strength of the material in a simple tension test, from the material certificate or standard.
- 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
- Both safety factors are this strength divided by the equivalent stress.
What this means: Required safety factor
- What it is
- The least factor you need between the yield strength and the von Mises equivalent stress, from your design code or rules.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 20. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.5 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
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.
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.
Method explained here · BIZOMAT original book
Yield under combined stress
In Yield and Plastic Flow by the Numbers
What it covers: Why shear, not pressure, makes metals yield, how the Tresca and von Mises criteria compare, and how to check the HC-63 barrel's bore without dropping the radial stress.
BIZOMAT tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.
Open the Triaxial Stress Check (Principal Stresses, von Mises, Tresca, Strains) calculator