Estimate Weibull parameters (β, η) from failure time data, and compute B10 life, MTTF, reliability function, and hazard rate in real time. Product life design and reliability analysis tool.
Data Input
Preset
Failure Time DataComma-separated (max 20 pts)
Unit: hours / days / cycles (arbitrary)
Time Unit
Estimation Results
Enter data and run analysis.
Results
—
Shape Parameter β
—
Scale Parameter η
—
B10 Life
—
MTTF
—
B50 Life (Median)
—
R(η) = 63.2% Point
—
Failure Mode
Weibull
Theory & Key Formulas
Weibull distribution PDF, CDF, and reliability function:
MTTF: $\eta\,\Gamma\!\left(1+\dfrac{1}{\beta}\right)$, Median rank: $F_i=\dfrac{i-0.3}{n+0.4}$
What is Weibull Analysis?
🙋
What exactly is Weibull analysis used for? I see it mentioned with reliability and failure data.
🎓
Basically, it's a powerful statistical method to model how long things last before they fail. It's not just for when things break, but when and how they break. In practice, you feed it failure times—like when lightbulbs burn out or bearings wear out—and it gives you two key parameters: shape (β) and scale (η). Try pasting some sample failure times into the simulator's data field above to see it in action.
🙋
Wait, really? So the shape parameter (β) tells me how things fail? What does a high or low β mean?
🎓
Exactly! β is incredibly insightful. If β < 1, failures decrease over time (like "infant mortality" in electronics). If β = 1, the failure rate is constant (random failures). If β > 1, failures increase over time due to wear-out, like an aging machine. For instance, a ball bearing might have a β of 2.5, showing clear wear-out behavior. Slide the β parameter in the simulator and watch how the hazard rate curve changes shape.
🙋
That makes sense! So what's the B10 life that the calculator shows? And how do we get that from the data?
🎓
Great question. B10 life is a crucial engineering metric—it's the time by which 10% of the population is expected to have failed. It's a conservative measure for design. The simulator calculates it directly once it estimates β and η from your data. A common case is in automotive: a warranty might be set just beyond the B10 life for a component. Change the failure data in the tool and see how the B10 life, MTTF, and reliability curve update in real-time.
Physical Model & Key Equations
The core of Weibull analysis is the reliability function, R(t), which gives the probability that an item will survive beyond a given time t. It is defined by the scale parameter η (characteristic life) and the shape parameter β (Weibull slope).
R(t): Reliability at time t (probability of survival). t: Time or number of cycles. η (eta): Scale parameter. The time at which approximately 63.2% of units have failed. β (beta): Shape parameter. Determines the failure rate trend (β < 1: decreasing, β = 1: constant, β > 1: increasing).
The hazard rate function, h(t), is the instantaneous failure rate at time t, given survival up to that time. It shows how the risk of failure changes over the component's life.
h(t): Hazard rate or instantaneous failure rate.
The equation shows that when β > 1, the hazard rate increases with time (wear-out), which is a common pattern for mechanical components like bearings and gears.
Real-World Applications
Fatigue Life Analysis from CAE/FEM: After running thousands of virtual fatigue simulations on a crankshaft, engineers get a distribution of failure cycles. Weibull analysis fits this data to predict the B10 life and reliability curve for the entire population, directly linking CAE results to a statistical warranty prediction.
Accelerated Life Testing (ALT): To predict the 10-year life of a new polymer seal, companies test it under high stress and temperature for a shorter time. Weibull analysis of this accelerated failure data allows extrapolation to normal use conditions, estimating the scale parameter η and shape parameter β for the product's lifetime.
Rolling Bearing Life Calculation (ISO 281): The standard L10 life (the life at which 90% of bearings survive) is a specific case of the Weibull B10 life. Bearing manufacturers use Weibull analysis with a fixed β (often ~1.5 for rolling contact fatigue) to rate their bearings and create the load-life curves you see in catalogs.
FMEA & Reliability Block Diagrams (RBD): In system reliability modeling, the failure rate for individual components in an FMEA or RBD is often not constant. The hazard rate function h(t) from Weibull analysis provides a time-dependent failure rate input, making system reliability predictions much more accurate over the product's lifecycle.
Common Misconceptions and Points of Caution
Weibull analysis is powerful, but it has several pitfalls. First, there is the misconception that "if you just input data, the answer magically appears." For example, a B10 life estimated from just 5 failure data points will have a very wide confidence interval, making it nearly useless in practice. You ideally want at least 20-30 data points or more. Next, there is the preconception that "the β value alone determines the failure cause." Before concluding "it's a wear-out failure!" just because β>3, you must cross-check it with the product's physical failure mechanism (e.g., metal fatigue, insulation degradation). β only indicates the statistical "shape"; identifying the root cause requires separate investigation.
Also, it's easy to forget about handling censored data (non-failure data). In a life test where only 20 out of 100 units have failed, the remaining 80 units provide the valuable information that they "have not failed yet." Note that methods using only the median rank method, like in this tool, cannot account for this censored data, leading to optimistic estimates (life tends to be estimated longer than it actually is) in such cases. In practice, the maximum likelihood estimation method, which can handle censoring, is often used.