BIZOMAT MAKEOS · Strength
Principal Stresses and von Mises
What are the principal stresses, maximum shear and von Mises stress for this stress state?
Principal stresses, their direction, maximum shear stress, von Mises and Tresca equivalent stresses for a plane stress state, with safety factors against yield.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Normal stress σx | MPa | Normal stress in the X direction (tension positive) from your calculation. |
| Normal stress σy | MPa | Normal stress in the Y direction (tension positive). |
| Shear stress τxy | MPa | Shear stress on the X and Y faces. |
| Yield strength | MPa | Minimum yield strength from the material certificate or specification, not a typical value. |
| Required safety factor | Factor of safety required by your design code, customer specification or company standard. |
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: Normal stress σx
- What it is
- The normal stress acting in the X direction at the point being checked, from your stress calculation. Tension is positive and compression negative.
- 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 120 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Include stress concentration effects before entering it; the tool does not add them.
What this means: Normal stress σy
- What it is
- The normal stress acting in the Y direction at the same point, at right angles to σx. Tension is positive and compression negative.
- 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
- Keep the sign: a compressive stress entered as positive changes every result.
What this means: Shear stress τxy
- What it is
- The shear stress acting on the X and Y faces of the same small element, for example from torsion or transverse shear.
- 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 50 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Its sign affects the principal angle but not the sizes of the principal stresses.
What this means: Yield strength
- What it is
- The minimum yield strength of the material at the point being checked, from the material certificate or specification. The equivalent stresses are compared with it.
- 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.
- Get it right
- Use the specified minimum, not a typical value.
What this means: Required safety factor
- What it is
- The minimum ratio of yield strength to equivalent stress the part must achieve, from your design code, customer specification or company standard.
- 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.
- Get it right
- This is a requirement you set; the tool only compares the calculated factor with it.
Method and formulas
- σ₁,₂ = (σx + σy) ÷ 2 ± √(((σx − σy) ÷ 2)² + τxy²); θp = ½ atan2(2τxy, σx − σy).
- von Mises σ′ = √(σ₁² − σ₁σ₂ + σ₂²); Tresca = max(|σ₁ − σ₂|, |σ₁|, |σ₂|) with σ₃ = 0.
- Safety factor = yield ÷ equivalent stress.
- Example: σx 120, σy -40, τ 50 MPa give σ₁ = 134.34 and σ₂ = -54.34 MPa, von Mises 168.23 MPa.
Limitations
- Plane stress (σz = 0); ductile, isotropic materials for the yield criteria.
- Stress concentrations must already be included in the entered stresses.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Combine bending, axial and torsional stresses at a critical point
- Check a hand calculation or FEA result
- Compare yield criteria
Who it is for
- Design engineers
- Structural and mechanical engineers
- Engineering students
What you get
- Principal stresses σ1 and σ2
- Principal angle
- Maximum shear stress
- von Mises and Tresca equivalent stresses
- Safety factors
Interpreting the result
- Use von Mises for ductile metals; Tresca is slightly more conservative.
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.
Explains the concept · BIZOMAT original book
Stress cycles: amplitude, mean and ratio
In Metal Fatigue by the Numbers
What it covers: How to describe a repeated stress by its amplitude, mean and ratio, find the nominal stresses in a shaft and a lever, and avoid losing the sign of a stress in a von Mises check.
Method explained here · BIZOMAT original book
Read stress results without fooling yourself
In Design Simulation by the Numbers
What it covers: Choose von Mises, principal or Tresca stress for the question, benchmark the solver, read averaged and unaveraged results and judge margins against the criteria.
Method explained here · BIZOMAT original book
Shear, torque and rosettes
In Strain Gauges and Load Cells by the Numbers
What it covers: Why gauges at 45° measure shear, how torque transducers work on a shaft, and how a three-gauge rosette gives the principal strains and stresses on a surface whose stress directions are unknown.
Method explained here · BIZOMAT original book
Principal stresses and Mohr's circle
In Stress Transformation by the Numbers
What it covers: How to find the largest and smallest normal stresses at a point, their planes and the largest shear, by formula, Mohr's circle and the principal stress tool, with the angle right.
Useful alongside · 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.
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