B10 Life⚠ unverified

Understanding the Parameters

• Scale parameter $\eta$ — the characteristic life, setting the timescale; $B_{10}$ is $\eta$ times the factor $(-\ln 0.9)^{1/\beta}$. Since $-\ln 0.9 \approx 0.105 \ll 1$, the B10 life is much shorter than the characteristic life — as it must be, since only $10\%$ have failed by B10 versus $63.2\%$ by $\eta$.

• Shape parameter $\beta$ — controls how the low-percentile factor behaves. For $\beta = 1$, $B_{10} = \eta(-\ln 0.9) \approx 0.105\,\eta$. Higher $\beta$ (tighter wear-out distribution) raises the factor toward $1$, so the early-failure life moves closer to the characteristic life — a tight distribution has few very early failures.

• The output $B_{10}$ — the design life at $90\%$ reliability. It is deliberately conservative: for a fleet, $10\%$ will have failed by $B_{10}$, so it bounds the onset of significant failures. Related metrics use other percentiles (B1 for $1\%$, B50 = median).

• The L10 bearing standard. For bearings the same quantity is the L10 life — the basic rating life at which $10\%$ fail — the foundation of the ISO 281 bearing life equation (Bearing Rating Life L10). Selecting a $10\%$ (not $50\%$) reference reflects that bearing failures should be rare in service.

Derivation (Approaching a Proof)

The B10 life is the $10\%$ percentile of the failure distribution — the time at which the cumulative failure probability reaches $0.10$, i.e. reliability falls to $0.90$. Set the Weibull survival function (Weibull Reliability) equal to $0.90$:

$$R(B_{10}) = e^{-(B_{10}/\eta)^\beta} = 0.90.$$

Take logarithms:

$$-\left(\frac{B_{10}}{\eta}\right)^\beta = \ln 0.90 \quad\Longrightarrow\quad \left(\frac{B_{10}}{\eta}\right)^\beta = -\ln 0.90.$$

Raise to the power $1/\beta$:

$$B_{10} = \eta\,(-\ln 0.90)^{1/\beta}. \qquad\blacksquare$$

This is the general B-life (percentile) formula $t_p = \eta\,[-\ln(1-p)]^{1/\beta}$ evaluated at $p = 0.10$; the median is the same with $p = 0.50$. All Weibull life percentiles come from inverting the survival function at the desired reliability — the B10 simply chooses the $90\%$-reliability point.

Dimensional check. $(-\ln 0.9)^{1/\beta}$ is a dimensionless number; $\eta$ carries the time units, so $B_{10}$ is a time. $\checkmark$

History and Development

• Bearing rating life. The B10/L10 concept originates with Arvid Palmgren and Gustaf Lundberg at SKF, whose 1940s work established the statistical basis of rolling-bearing life and the $10\%$-failure rating life that became ISO 281 (Bearing Rating Life L10, Bearing Life hours). Bearings are inherently Weibull-distributed, with characteristic $\beta \approx 1.1$–$1.5$.

• Why $10\%$, not $50\%$. Choosing a low percentile as the design reference reflects a design philosophy: for load-bearing mechanical components, the acceptable failure fraction is small, so the early life — not the average — governs. This conservatism is baked into machine-design practice.

• The B-life family. Different industries adopt different percentiles (B10, B1, B0.1 for very safety-critical parts), all computed from the same Weibull percentile inversion, tuning the acceptable failure fraction to the consequences of failure.

Related Concepts: Weibull Median, Weibull Mean, Weibull Mode, Weibull Reliability, Bearing Rating Life L10, Bearing Life hours, Weibull Distribution PDF

Notes: Registry calculator b10-life (unverified). $B_{10} = \eta(-\ln 0.9)^{1/\beta}$ — correct as shipped; the $10\%$-failure ($90\%$-reliability) life, = bearing L10. Much shorter than $\eta$ (the $63.2\%$ point). Special case of $t_p = \eta[-\ln(1-p)]^{1/\beta}$. Registry labels $\eta$ dimensionless (is a time).