Factor Of Safety Ultimate⚠ unverified
Mechanical / Stress Analysis · Compute the factor of safety against ultimate (fracture) failure
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| Sut | Sut | Pa | 1.0 | Material ultimate tensile strength |
| sigma_max | σmax | Pa | 1.0 | Maximum applied stress |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | n | — | Dimensionless factor of safety against ultimate failure |
The science & history
Understanding the Parameters
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Ultimate strength $S_{ut}$ — the highest stress a tension specimen sustains before rupture. It is the capacity for a fracture check, always higher than the yield strength for ductile metals.
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Maximum applied stress $\sigma_{max}$ — the peak stress in the part. For brittle materials this is usually the maximum principal stress (brittle fracture follows the maximum-normal-stress theory); for a stress raiser, use the peak $K_t\sigma_{nom}$ (Stress Concentration) since brittle materials do not redistribute stress by yielding.
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Factor $n$ — the margin against fracture. For brittle materials it is the primary safety measure (there is no yield plateau to fall back on), and large values are used because brittle failure is sudden and without warning.
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Ductile vs brittle — ductile parts are normally sized on yield (smaller, governing margin), with the ultimate margin as a secondary check; brittle parts are sized on ultimate directly.
Derivation (Approaching a Proof)
Fracture is taken to occur when the maximum stress reaches the ultimate strength:
$$\sigma_{max} = S_{ut} \quad\text{(fracture)}.$$
The factor of safety is the ratio of that capacity to the applied peak stress — equivalently, the factor by which the (proportional) load may be scaled before fracture:
$$n = \frac{S_{ut}}{\sigma_{max}}.$$
At $n = 1$ the peak stress equals the ultimate strength; $n > 1$ leaves margin. For brittle materials, using the maximum principal stress for $\sigma_{max}$ makes this the maximum-normal-stress (Rankine) failure check.
Dimensional check. $n = \dfrac{S_{ut}}{\sigma_{max}} = \dfrac{\text{Pa}}{\text{Pa}}$ = dimensionless — a pure factor, as required.
History and Development
The maximum-normal-stress theory pairing with an ultimate-strength margin is the classical Rankine criterion (W.J.M. Rankine, 19th century), appropriate for brittle materials such as cast iron, ceramics, and glass that fracture on the plane of maximum tensile stress. The distinction between yield-based and ultimate-based factors of safety — and choosing the right one for the material's failure mode — is a cornerstone of design practice.
Related Concepts: Factor of Safety yield, Max Principal Stress, Stress Concentration, Principal Stresses, Static Failure Theories, Von Mises Stress
Notes: Guards fracture (use for brittle materials); ductile parts usually governed by yield (Factor of Safety yield). For brittle/stress-raiser cases use the peak principal stress $K_t\sigma_{nom}$ (brittle materials don't redistribute). Maximum-normal-stress (Rankine) theory.