Weibull Reliability Calculator (R(t), Failure Rate, MTTF, B10)
By Santaji GadepatilinUpdated November 8, 2026
The Weibull distribution is the standard model for component life because its shape parameter β covers early failures (β < 1), random failures (β = 1) and wear-out (β > 1). This Weibull reliability calculator gives the reliability and probability of failure at a chosen time, the instantaneous failure rate, the mean time to failure and the B10 life from β and the characteristic life η.
How to use the Weibull reliability calculator
Enter the shape parameter β and the characteristic life η (from a Weibull plot of failure data or a supplier’s life test) and the operating time t. The calculator returns the reliability and probability of failure at t, the hazard rate at t in failures per million hours, the mean life and the B10 life. To find the time for a target reliability, enter R and press × on t.
Weibull reliability formulas
R(t) = e−(t/η)β · F(t) = 1 − R(t)
h(t) = (β/η)(t/η)β−1 · MTTF = η Γ(1 + 1/β) · B10 = η (−ln 0.9)1/β
Worked example
A pump seal with β = 1.5 and η = 10,000 h, after t = 5,000 h:
- (t/η)β = 0.51.5 = 0.3536, so R = e−0.3536 = 70.22% and F = 29.78%
- Hazard rate = (1.5 ÷ 10,000) × 0.50.5 = 106.1 per 10⁶ h — and rising, because β > 1
- MTTF = 10,000 × Γ(1.667) = 9,027 h; B10 life = 2,231 h
Weibull reliability at multiples of η
| β | t = 0.1η | t = 0.5η | t = η | t = 2η | MTTF ÷ η | B10 ÷ η |
|---|---|---|---|---|---|---|
| 0.5 | 72.9% | 49.3% | 36.8% | 24.3% | 2.0000 | 0.0111 |
| 1 | 90.5% | 60.7% | 36.8% | 13.5% | 1.0000 | 0.1054 |
| 1.5 | 96.9% | 70.2% | 36.8% | 5.9% | 0.9027 | 0.2231 |
| 2 | 99.0% | 77.9% | 36.8% | 1.8% | 0.8862 | 0.3246 |
| 3.5 | 100.0% | 91.5% | 36.8% | 0.0% | 0.8997 | 0.5257 |
Practical notes
- With β > 1, preventive replacement before wear-out makes sense; with β ≤ 1 it does not — replacing a part early only resets the infant-mortality period.
- Fit β and η from field data with a Weibull probability plot (median ranks) or maximum likelihood; at least 5–10 failures give a usable β.
- A three-parameter Weibull adds a failure-free time γ: replace t with (t − γ) when no failures can occur before γ.
- For β = 1 the result equals the exponential model used for MTBF and SIL calculations, with η = MTBF.
Frequently asked questions
What is the Weibull reliability function?
R(t) = exp(−(t/η)^β). At t = η the reliability is always e⁻¹ ≈ 36.8%, whatever β is — so η is the time by which 63.2% of units have failed.
What does the shape parameter β mean?
β < 1 means a falling failure rate (infant mortality), β = 1 a constant failure rate (the exponential distribution, random failures) and β > 1 a rising failure rate (wear-out). Bearings are typically around 1.1–1.5 and fatigue or corrosion failures 2–4.
How is MTTF calculated for a Weibull distribution?
MTTF = η × Γ(1 + 1/β), where Γ is the gamma function. For β = 1 it equals η; for β = 2 it is 0.886 η; for β = 3.5 it is about 0.9 η.
What is B10 life?
The time by which 10% of the population has failed (90% reliability), B10 = η × (−ln 0.9)^(1/β). It is the usual rating life for rolling bearings.