Library / Mechanical / MEC-57
Mechanical
Yield utilisation against a required factor of safety
Fraction of the yield strength used up once the required factor of safety is applied to the equivalent stress.
- Ref
- MEC-57
- Rev
- v1
- Field
- Mechanical
- Result
- U
U is the fraction of the yield strength used once the required factor of safety nreq is applied to the equivalent stress σeq (from Tresca or von Mises). U ≤ 1 means the specified margin is met and U > 1 means it is not. It is the inverse view of the factor of safety: the achieved factor is n = Sy/σeq, and U = nreq/n. Zero equivalent stress means there is no yield demand under that criterion, not that every failure mechanism is excluded. A factor-of-safety ratio is a load multiplier only where the stress scales in proportion to the load.
Fraction of the yield strength used up once the required factor of safety is applied to the equivalent stress. A value of 1 or less means the specified margin is met.
Calculator
- Factor of safety achieved, Sy/σeq2.5
- Margin remaining, Sy − nreqσeq50 MPa
- Largest equivalent stress allowed, Sy/nreq125 MPa
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- With the lecture values, Tresca 100 MPa and von Mises 88.88 MPa against a yield strength of 250 MPa and a required factor of 2, utilisation is 0.8 for Tresca and 0.711 for von Mises. Both are below 1, so both pass, with the von Mises result less severe.
- At the defaults (σeq = 100 MPa) the achieved factor of safety is 2.5 against a requirement of 2, so 20 % of the required capacity is spare.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| σeq Equivalent stress (Tresca or von Mises) | 0 to 500 MPa | 0 | 4 | 4 | +10% | +10 % |
| Sy Uniaxial yield strength | 50 to 1,500 MPa | 0.1333 | 4 | 3.867 | +10% | −9.09 % |
| nreq Required factor of safety | 1 to 5 | 0.4 | 2 | 1.6 | +10% | +10 % |
Sweep of Equivalent stress (Tresca or von Mises)
| Equivalent stress (Tresca or von Mises) (MPa) | Utilisation |
|---|---|
| 0 | 0 |
| 62.5 | 0.5 |
| 125 | 1 |
| 187.5 | 1.5 |
| 250 | 2 |
| 312.5 | 2.5 |
| 375 | 3 |
| 437.5 | 3.5 |
| 500 | 4 |
Sweep one variable
Variables and how each one contributes
Utilisation
What the equation gives you. Everything else feeds it.
Required factor of safety
The factor of safety the design requires, a plain number. Typical values lie between 1.5 and 3 depending on how well the loads and material are known. Raises U in direct proportion. Raising it by 10 % (to 2.2) moves the result by +10 %, from 0.8 to 0.88.
Equivalent stress (Tresca or von Mises)
The equivalent stress from the Tresca or von Mises calculation, in MPa. It is zero or positive. Raises U in direct proportion. Raising it by 10 % (to 110) moves the result by +10 %, from 0.8 to 0.88.
Uniaxial yield strength
The uniaxial yield strength of the material, in MPa. It must be positive. Lowers U in inverse proportion: a stronger material uses a smaller fraction of its strength. Raising it by 10 % (to 275) moves the result by −9.09 %, from 0.8 to 0.7273.
Worked example with the values above
- Required stress capacity: nreq σeq = 2 × 100 = 200 MPa.
- Divide by the yield strength: 200 / 250.
- U = 0.8, so the specified margin is met.
Assumptions and limits
- The equivalent stress comes from a criterion suited to the material, usually Tresca or von Mises for ductile metals.
- Yield is the limit that matters; fatigue, buckling and brittle fracture are separate checks.
Common mistakes
- Dividing by the ultimate tensile strength when yield is the limit.
- Using a negative or zero yield strength or required factor of safety.
Statistics and uncertainty
A single equivalent stress hides scatter in loads, geometry and material. If the stress and strength each vary by 10 % (one standard deviation), U varies by about 14 % of its value, so a utilisation of 0.9 with nreq = 2 leaves little room, while 0.6 leaves a comfortable one.
Where it is used
- Turning a Tresca or von Mises result into a pass or fail.
- Comparing the margin of several load cases in one table.
Maths you need
- Yield strength: onset of permanent deformation
- Factor of safety: strength divided by working stress