Size Factor⚠ unverified
Mechanical / Fatigue · Compute the size modification factor kb (Shigley Eq. 6-20)
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| d | d | mm | 1.0 | Characteristic diameter, in millimetres (mm), for bending or torsion |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | kb | — | Size modification factor kb (dimensionless) |
The science & history
Understanding the Parameters
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Characteristic diameter $d$ — for a round rotating shaft it is the actual diameter; for non-rotating or non-round sections an effective diameter is defined from the equally-stressed volume. It is the sole driver of the real $k_b$.
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Why size lowers endurance — fatigue cracks start at the surface where stress is highest. A larger part has more surface area and, under bending/torsion, a shallower stress gradient, so a larger volume sits near peak stress — statistically more likely to contain a critical flaw. Hence $k_b < 1$ and decreasing with $d$.
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Bending/torsion vs axial — the effect comes from the stress gradient, which exists only in bending and torsion. Under uniform axial stress there is no gradient and $k_b = 1$ (the registry constant is right for that case, wrong for bending/torsion).
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Magnitude — modest but real: $k_b$ ranges from $1$ at a few mm down to $\sim 0.7$ for large shafts.
Derivation (Approaching a Proof)
The size factor is empirical, fitted to rotating-beam data across specimen diameters. The mechanism is a weakest-link / stressed-volume argument: model the surface as containing randomly distributed initiation sites; the probability that a critical flaw lies in the highly-stressed region grows with that region's volume. Under bending, the fraction of the section stressed above $95\%$ of the surface value (Shigley's "$A_{95}$" stressed area) scales with $d^2$, so larger parts expose proportionally more vulnerable material. Fitting endurance to diameter yields the power laws
$$k_b = \left(\frac{d}{7.62}\right)^{-0.107}, \quad 2.79 \le d \le 51\ \text{mm},$$
(with $d$ in mm), and a second exponent for larger diameters. For axial loading the stressed volume is the whole section regardless of size, removing the gradient dependence, so $k_b = 1$.
Dimensional check. $k_b$ is dimensionless: in the power-law form $d$ is normalised by a reference diameter ($7.62\ \text{mm}$) before exponentiation, so the ratio and its power are pure numbers.
History and Development
The size factor is one of the six Marin factors introduced by Joseph Marin and codified in Shigley's Mechanical Engineering Design, which decompose the gap between the idealised specimen endurance limit $S_e'$ and a real part's $S_e$ into multiplicative corrections. The size effect was among the earliest fatigue observations to resist a purely deterministic explanation, motivating the statistical weakest-link theories (Weibull) of fatigue scatter.
Related Concepts: Marin Endurance Limit, Marin Modification Factors, Temperature Factor, Reliability Factor, Endurance Limit steel, Fatigue Failure Variable Loading
Notes: Registry returns constant $k_b = 1$ regardless of $d$ (placeholder) — use Shigley Eq. 6-20 for bending/torsion. $k_b = 1$ is correct for axial loading (no stress gradient). Needs an effective diameter for non-round/non-rotating sections.