Life Reliability Factor⚠ unverified
Mechanical / Bearings · Compute the reliability life adjustment factor a1 (Weibull)
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| reliability | reliability | — | 1.0 | Desired reliability as a fraction between 0 and 1 (dimensionless) |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | a1 | — | Reliability life adjustment factor a1 (dimensionless). Returns 1.0 for reliabilities of 0.9 or below |
The science & history
Understanding the Parameters
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Reliability $R$ — the fraction of a large bearing population expected to survive the adjusted life, entered as a decimal (0.90, 0.95, 0.99, 0.999…). It is the complement of the failure probability: $R = 0.99$ means 1 % are allowed to fail. As $R \to 1$ the demanded life shrinks toward zero, because insisting that virtually none fail forces you to quote a life so short that almost the whole batch is still intact.
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The factor $a_1$ — the multiplier on $L_{10}$. It is 1 at the reference $R = 0.90$ and drops steeply as $R$ rises: roughly $a_1 \approx 0.62$ at 95 %, $\approx 0.44$ at 96 %, $\approx 0.21$ at 99 %. High reliability is expensive in life — a small increase in required survival can halve the usable life.
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The constants $0.87$, $0.67$, $0.27$ — empirical coefficients that shape the Weibull-derived curve so it passes through $a_1 = 1$ at $R = 0.90$ and matches ISO 281's tabulated values across the practical 90–99.95 % range. They are a curve-fit, not fundamental physics.
Derivation (Approaching a Proof)
Bearing fatigue life follows a two-parameter Weibull distribution: the probability that a bearing survives to life $L$ is
$$R(L) = \exp\!\left[-\left(\frac{L}{L_c}\right)^{\!\beta}\right],$$
where $\beta$ is the Weibull slope (dispersion of the fatigue data, $\approx 1.5$ for bearings) and $L_c$ is a characteristic life. Inverting for the life at a given reliability $R$:
$$L(R) = L_c \left(\ln\frac{1}{R}\right)^{1/\beta}.$$
Form the ratio to the reference 90 % life $L_{10} = L(0.90)$:
$$a_1 = \frac{L(R)}{L_{10}} = \left(\frac{\ln(1/R)}{\ln(1/0.90)}\right)^{1/\beta}.$$
So $a_1$ is fundamentally $\big(\ln(1/R)\big)^{1/\beta}$ normalised at $R=0.90$. The registry's closed form $\left(0.87\,\ln(1/R)\right)^{0.67} + 0.27$ is the ISO-standard empirical fit to exactly this Weibull ratio: the exponent $0.67 \approx 1/\beta$ recovers the Weibull slope $\beta \approx 1.5$, the coefficient $0.87$ and the additive $0.27$ nudge the curve to hug the tabulated ISO values over the whole reliability range (the pure power law and the fitted form agree closely near 90 % and the fit tracks the data better in the extreme-reliability tail).
Dimensional check. $R$ is a dimensionless probability, so $a_1$ is dimensionless. ✓
History and Development
The reliability adjustment entered the standards as the $a_1$ factor in ISO 281 and the AFBMA/ANSI adjusted-rating-life framework of the 1970s–80s. It rests directly on Weibull's statistical theory of the strength of materials (Waloddi Weibull, 1939, 1951) applied to the Lundberg–Palmgren fatigue model. The specific fitted expression used here matches the $a_1$ column tabulated in ISO 281 for reliabilities from 90 % ($a_1 = 1$) up to 99.95 %. It is the first of the three modifiers combined in Bearing Life Modifier.
Related Concepts: Bearing Rating Life L10, Bearing Life Modifier, Weibull Reliability, Weibull Failure Rate, Weibull Distribution PDF
Notes: Enter $R$ as a fraction (0.99, not 99). Below $R = 0.90$ the factor is clamped to 1.0 — the standard does not extend the adjustment to reliabilities lower than the reference.