Bathtub Curve (Weibull Hazard Rate)
Enter Weibull shape parameter β and scale parameter η to plot the hazard rate h(t) curve, identify infant mortality, random failure, and wear-out phases, and get maintenance recommendations.
Weibull Parameter Settings
β<1: Infant mortality · β=1: Random · β>1: Wear-out
63.2% of units fail before η hours
Weibull Hazard Rate Formula
h(t) = (β / η) × (t / η)β − 1
= (2.5 / 1,000) × (t / 1,000)1.5
Hazard Rate Curve h(t) with Three Phases
x-axis divided into three equal phases · Left: Infant Mortality (orange) · Middle: Random Failures (blue) · Right: Wear-out (red)
Current phase:Wear-out
Failure rate increases over time (β = 2.5 > 1). Equipment failure probability rises continuously due to fatigue, wear, corrosion, or aging. Higher β means steeper increase.
Recommendation: Set up a preventive maintenance (PM) schedule based on η (characteristic life). Schedule periodic part replacement or overhaul before failure rate rises significantly. Consider RCM (Reliability-Centered Maintenance) analysis.
Three Phases of the Bathtub Curve
Infant Mortality (β < 1)
Decreasing failure rate. Common with design defects, material flaws, or poor manufacturing. Burn-in screening eliminates weak units so shipped products operate in the stable zone.
Random Failures (β = 1)
Constant failure rate following an exponential distribution (memoryless). MTBF = η; failures are purely random, age-independent. This is the longest phase — the useful service life.
Wear-out (β > 1)
Increasing failure rate. Accelerated failure from fatigue, corrosion, or material aging. Establish a PM schedule before characteristic life η to avoid the high-risk zone.
Typical β Values
| β Range | Failure Mode | Typical Example |
|---|---|---|
| β ≈ 0.5 | Infant Mortality (severe) | Early defects in electronics, poor soldering |
| β ≈ 0.8 | Infant Mortality (mild) | Early break-in issues in mechanical parts |
| β = 1 | Random Failure | Random failures in resistors/capacitors, external shock |
| β ≈ 1.5 | Mild Wear | Early wear in rolling bearings |
| β ≈ 2.5 | Normal Wear | Mechanical drive components, shaft seal wear |
| β ≈ 4 | Severe Wear | Metal fatigue, severe gear wear |
| β > 5 | Rapid Aging | Rubber degradation, paint deterioration |
A reliability engineer needs to analyze the failure behavior of a new electronic control board across the infant-mortality, useful-life, and wear-out phases. The bathtub curve visualizes how the hazard rate changes over time via the Weibull distribution, guiding burn-in testing, maintenance intervals, and replacement strategy. This tool is especially useful for life-cycle cost analysis — deciding when to switch from a run-to-failure strategy to preventive replacement to minimize total maintenance cost.
Product life in three phases: Infant mortality (β=0.5, η=500 h): At t=100 h, hazard rate λ = β/(η×(t/η)^(1-β)) — rapidly decreasing trend → Recommend a 168-hour burn-in screen Random failures (β=1.0, η=5000 h): MTBF ≈ η = 5000 h, random failures Wear-out (β=3.0, η=8000 h): B10 life = η×(-ln(0.9))^(1/β) = 4200 h → Recommend preventive replacement at 4000 h
FAQ
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The random-failure phase of the bathtub curve (β≈1) follows an exponential distribution; its mean time between failures is the MTBF, computed directly with the MTBF tool.
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