Reliability Factor⚠ unverified

Understanding the Parameters

• Target reliability $R$ — the probability that a part survives to its design life (e.g. $0.99$ = 99 %). Higher reliability demands a larger derating (smaller $k_e$), because you must design further out on the weak tail of the endurance-limit distribution.

• Standard normal variate $z_a$ — the number of standard deviations below the mean corresponding to $R$: $z_a = 1.288$ for 90 %, $2.326$ for 99 %, $3.091$ for 99.9 %. It is what converts a reliability into a multiplier.

• The $0.08$ coefficient — the assumed coefficient of variation of the endurance limit ($\sim 8\%$), the empirical scatter of rotating-beam data. Each standard deviation costs $8\%$ of endurance.

• $k_e = 1$ at 50 % — designing to the mean endurance limit gives only a coin-flip chance of survival; every real design pushes $k_e$ below 1.

Derivation (Approaching a Proof)

Treat the specimen endurance limit as a normally distributed random variable with mean $\mu = S_e'$ and standard deviation $\sigma = 0.08\,\mu$ (an $8\%$ coefficient of variation from test scatter). To guarantee survival with probability $R$, design to the endurance value at the $R$-quantile of the lower tail:

$$S_{e,R} = \mu - z_a\,\sigma = \mu(1 - 0.08\,z_a),$$

where $z_a = \Phi^{-1}(R)$ is the standard-normal variate. The reliability factor is the ratio of this derated value to the mean:

$$k_e = \frac{S_{e,R}}{\mu} = 1 - 0.08\,z_a.$$

Evaluating $z_a$ for standard reliabilities produces the tabulated $k_e$ (e.g. $R = 0.99 \Rightarrow z_a = 2.326 \Rightarrow k_e = 1 - 0.08(2.326) = 0.814$).

Dimensional check. $k_e = 1 - 0.08\,z_a$ is dimensionless ($z_a$ is a pure standard-normal variate), as a modification factor must be.

History and Development

The reliability factor injects statistical fatigue into the otherwise deterministic Marin framework (Joseph Marin; Shigley). It formalises the recognition — traceable to the large fatigue-scatter datasets of the mid-20th century — that an endurance "limit" is really a distribution, and that safety must be stated probabilistically. The $8\%$ coefficient of variation and the normal-tail construction are the standard textbook simplification of that idea.

Related Concepts: Marin Endurance Limit, Marin Modification Factors, Size Factor, Temperature Factor, Endurance Limit steel, Fatigue Failure Variable Loading

Notes: Registry returns constant $k_e = 0.814$ (the 99 % value) regardless of input — use $k_e = 1 - 0.08\,z_a$ (or the table) for other reliabilities. Assumes an $8\%$ coefficient of variation and a normal distribution. $k_e = 1$ at 50 % (design to the mean).