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Weibull analysis

The Weibull distribution is the most widely used model for product lifetime data in reliability engineering. Its flexibility—provided by two parameters—lets it fit early-life failures, constant-hazard failures, and wear-out failures.

The two-parameter Weibull

Reliability function: R(t) = e^(−(t/η)^β) Failure rate: h(t) = (β/η) × (t/η)^(β−1) β = shape parameter (dimensionless) η = scale parameter = characteristic life (same units as t)

At t = η: R(η) = e⁻¹ = 0.368, regardless of β. The characteristic life η is always the time at which 63.2% of units have failed.

Interpreting β

β value	Failure pattern	Hazard rate	Bathtub region
β < 1	Early-life (infant mortality)	Decreasing	Left side
β = 1	Random (exponential)	Constant	Bottom (flat)
β > 1	Wear-out	Increasing	Right side

Worked example

A bearing has β = 2.5 and η = 1000 h. Find R(500).

R(500) = e^(−(500/1000)^2.5) = e^(−(0.5)^2.5) (0.5)^2.5 = 0.1768 R(500) = e^(−0.1768) = 0.8380

Interpretation: 83.8% of bearings survive past 500 hours. The β = 2.5 > 1 confirms this is a wear-out mode (the failure rate increases over time).

Finding B10 life (time at 10% failure)

Set R(t) = 0.90 and solve for t: 0.90 = e^(−(t/1000)^2.5) ln(0.90) = −0.10536 (t/1000)^2.5 = 0.10536 t/1000 = 0.10536^(1/2.5) = 0.10536^0.4 ≈ 0.4065 B10 life ≈ 406 h

Common exam traps

η is NOT the mean life (MTTF). MTTF depends on both β and η: MTTF = η × Γ(1 + 1/β). Only when β = 1 does MTTF = η.
B10 is the 10% failure point, not the 10% reliability point. B10 is defined as the time at which 10% of units have failed (R = 0.90).
Weibull paper is log-log. Plotting cumulative failure % on Weibull probability paper linearizes the CDF; the slope gives β and the x-intercept at 63.2% gives η.

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