The bathtub curve is one of six: failure patterns, Weibull and the reliability block diagram
The bathtub curve fits a few per cent of items. The six patterns the airline studies found, what each implies for the kind of task, how failure rate relates to MTBF, how the Weibull shape parameter reads the pattern from data, and how a reliability block diagram puts the parts together into a plant's availability.
Key takeaways
- The bathtub curve is the familiar picture of failure rate against age: high at first, flat in the middle, rising at the end. It is one of six patterns, and the airline studies that defined them found it fitted only a few per cent of items.
- Most complex items follow pattern F, high early then flat, or pattern E, flat throughout. For those, replacing a part at an age does nothing, and an intrusive task introduces the failures it was meant to prevent.
- The pattern decides the kind of task. Wear-out earns a fixed interval. Random earns condition monitoring or failure finding. Infant mortality earns leaving working equipment alone.
- The failure rate is a property of a population. The Weibull shape parameter reads it from failure data: below 1 is early failure, about 1 is random, above 1 is wear-out.
- A reliability block diagram puts the items together: series for anything that must all work, parallel for redundancy, so the plant's availability can be read from its parts.
Everyone who has sat through a reliability course remembers the bathtub curve. Failure rate high at the start, flat through the middle, rising at the end. It is a good picture of a light bulb and a poor picture of a gearbox, and the maintenance programs built on it are why so many plants replace parts on a schedule that does not change how often they fail.
The correction came from the airline industry in 1978, when a study of the failure history of aircraft components found six patterns rather than one, and found that the bathtub fitted very few items. That finding founded reliability centred maintenance and changed what a preventive task is for. This guide sets out the six patterns and what each implies for the task, how failure rate relates to the measures a plant already uses, how the Weibull distribution reads the pattern from data, and how a reliability block diagram puts the parts together into a plant.
What the bathtub curve says, and what it does not
The curve plots the conditional probability of failure, the chance an item that has survived to a given age fails in the next period, against age. Its three regions are infant mortality, where manufacturing and installation defects fail early; useful life, where failures are random and the rate is flat; and wear-out, where an ageing mechanism takes over and the rate climbs. The implication, if it holds, is a burn-in to clear the early failures and a replacement age short of the wear-out.
It holds for simple items with one dominant ageing mechanism and little else: a bearing in a clean, well-aligned machine, a tyre, a lamp. It does not hold for a pump, a drive, a controller or a gearbox, because those have many parts and many failure modes, and the sum of their patterns is not a bathtub. That is what the airline study found when it looked.
The six patterns
Nowlan and Heap's 1978 report for United Airlines classified the failure behaviour of aircraft components into the six patterns in the figure. The shares are from that study and later studies in other industries found similar shapes with different proportions. What every one of them found is the same: the patterns with no wear-out age, D, E and F, cover most items, and the infant mortality pattern F on its own covers more than half.
The practical reading is in the table. Only A, B and C have a rising rate to aim a replacement at, and only B has a clear enough knee to make a fixed interval efficient. For everything else an age-based task cannot help, and for pattern F it can hurt, because an intrusive task returns the item to the high early end of its curve. That is the finding behind the rule in the preventive maintenance guide that the kind of task is chosen from how the failure behaves.
ABathtub4 per cent
Failure rate against age
Burn-in to clear the early failures, then an age limit short of the wear-out
BWear-out2 per cent
Failure rate against age
A fixed interval replacement short of the wear-out age earns its place
CGradual increase5 per cent
Failure rate against age
Condition inspection on the climb, not a replacement age, because there is no knee to aim at
DInitial increase7 per cent
Failure rate against age
No age limit helps. Condition monitoring if the failure gives warning
ERandom14 per cent
Failure rate against age
Replacement at an age does nothing. Condition tasks or failure finding, or accept and repair
FInfant mortality68 per cent
Failure rate against age
An intrusive task introduces the failures it was meant to prevent. Leave what is working alone, and monitor
Shares from the 1978 United Airlines study. Most items show no wear-out age to aim a replacement at
| Pattern | Shape | Typical of | What it implies for the task |
|---|---|---|---|
| A: Bathtub | High early, flat, then rising at the end | Simple items with a wear-out mode and some installation failures | Burn-in to clear the early failures, then an age limit short of the wear-out |
| B: Wear-out | Flat, then rising sharply at an age | Items with one dominant wear mechanism: liners, screen media, tyres, some seals | A fixed interval replacement short of the wear-out age earns its place |
| C: Gradual increase | Rising slowly with age, no clear wear-out point | Fatigue and corrosion, where the chance of failure climbs but never jumps | Condition inspection on the climb, not a replacement age, because there is no knee to aim at |
| D: Initial increase | Low at first, rising quickly to a constant | Items that settle in before they can fail, then fail at random | No age limit helps. Condition monitoring if the failure gives warning |
| E: Random | Constant at every age | Electronics, hydraulics, many complex assemblies | Replacement at an age does nothing. Condition tasks or failure finding, or accept and repair |
| F: Infant mortality | High at first, falling to a constant | Complex items with electronics or many parts, and anything freshly maintained | An intrusive task introduces the failures it was meant to prevent. Leave what is working alone, and monitor |
An intrusive overhaul on equipment that follows pattern F does not reset the clock. It restarts the infant mortality.
Failure rate, MTBF and the measures a plant uses
Failure rate is a property of a population: failures per unit of operating time, read over a period, and the patterns are how it changes with age. MTBF is its inverse when the rate is constant, which is the pattern E assumption, and most plants quote an MTBF without asking whether the assumption holds. Two pumps with the same MTBF can be entirely different animals: one failing at random throughout its life, the other running clean for years and then failing in a cluster.
The difference matters for what to do. A random pattern says nothing about when the next failure will come, so the response is to detect it coming, through condition monitoring, or to test for it where it is hidden. A wear-out pattern says a good deal about when, so a replacement age or a wear measurement earns its place. The maintenance KPIs guide reads MTBF alongside the other measures for this reason: it is a number, not a strategy.
Reading the pattern from data: Weibull
The Weibull distribution is the tool that reads the pattern from real failure ages. Its shape parameter, beta, sets how the failure rate changes with age: below one and the rate falls, which is early-life failure; near one and it is constant; above one and it rises, with the steepness growing as beta does. Its scale parameter, eta, is the characteristic life, the age by which about 63 per cent of a population has failed. Fit a Weibull to a dozen or more failure ages for one failure mode and the two numbers say which pattern the item follows and roughly where its wear-out sits.
The Weibull calculator on this site takes the two parameters and gives reliability at an age, the B10 life and the conditional reliability of an item that has already survived. IEC 61649 is the standard for the analysis, and the one discipline it insists on is that the data be for one failure mode on one population. A Weibull fitted to every failure of a pump, bearings and seals and couplings together, produces a beta near one whatever the parts are doing, and says random when the bearings are plainly wearing out.
Putting the parts together: the reliability block diagram
A plant is many items, and what its owner cares about is the availability of the whole. The reliability block diagram, standardised in IEC 61078, puts the items together as blocks: in series where every item must work for the system to work, which is most of a processing train, and in parallel where any one of a group is enough, which is a duty and standby pair. The reliability or availability of a series chain is the product of its blocks, so a line of ten items each at 98 per cent sits at about 82 per cent, and the chain is weaker than its weakest link. Redundancy in parallel raises the figure, at the cost of the second item.
The diagram's value is in what it shows: which block the plant's availability actually depends on, which is usually not the most expensive one, and what a second unit or a spare would buy. It is the right tool for the critical spares decision on a single item, for the duty and standby argument on a pump, and for the first pass of a criticality assessment where the consequence of an item's failure is the production it stops.
| Arrangement | System availability | Example |
|---|---|---|
| Two items in series, each 98 per cent | About 96 per cent | A conveyor and the crusher it feeds |
| Ten items in series, each 98 per cent | About 82 per cent | A crushing and screening train |
| Two items in parallel, each 98 per cent, one needed | Above 99.9 per cent | Duty and standby lube pumps |
| A 98 per cent item protected by a 90 per cent standby | About 99.8 per cent | A main pump with an older spare that starts most of the time |
Where the reasoning goes wrong
- A bathtub assumed for everything, so every item gets an overhaul age and the infant mortality items fail after each one.
- MTBF quoted as if it were a strategy, with no idea of the pattern behind it.
- A Weibull fitted to mixed failure modes, which always reads as random.
- Fewer than a handful of failures fitted, and the parameters quoted to three decimal places.
- Vendor intervals treated as wear-out ages when the vendor's basis was the warranty, not the pattern.
- Availability targets set for a train without a block diagram, so nobody can say which item the target depends on.
- Redundancy installed without a failure-finding test on the standby, so the parallel block is a hope.
Common questions
What is the bathtub curve?
A plot of failure rate against age that is high in early life, falls to a flat floor, and rises again as the item wears out, so that it looks like a bath in section. The three regions are called infant mortality, useful life and wear-out. It is a useful teaching picture and a poor general model: the airline studies that classified real failure patterns found it fitted about four per cent of items.
What are the six failure patterns?
The patterns identified in the United Airlines study by Nowlan and Heap in 1978 that founded reliability centred maintenance. A is the bathtub. B is wear-out, flat then rising. C is a gradual increase with no clear wear-out point. D is a quick rise from low to a constant. E is constant at every age. F is high early then falling to a constant, called infant mortality. In that study, patterns D, E and F, which have no wear-out age, covered 89 per cent of items.
What does the bathtub curve mean for maintenance?
That an age-based replacement only helps where the failure rate rises with age, which is patterns A, B and C. For the random and infant mortality patterns that most complex equipment follows, replacing a part at an interval does not reduce failures and, because freshly maintained equipment sits at the high early end of pattern F, can increase them. Those failures are managed by condition monitoring where they give warning, failure-finding tests where they are hidden, and otherwise by accepting and repairing.
What is the difference between failure rate and MTBF?
Failure rate is the number of failures per unit of operating time for a population, and it can change with age, which is what the patterns describe. Mean time between failures is the average operating time between failures, and for a constant failure rate it is simply the inverse. MTBF on its own says nothing about the pattern: two items with the same MTBF can have very different ages at which they fail. The MTBF calculator on this site works the figure out from failures and hours.
What does the Weibull shape parameter tell you?
Which pattern the failure data follows. The Weibull distribution has a shape parameter, beta, that sets how the failure rate changes with age. Beta below 1 means the rate is falling, early-life failure. Beta near 1 means constant, random failure. Beta above 1 means rising, wear-out, and the higher it is the sharper the wear-out. Fitting a Weibull to a set of failure ages, as the Weibull calculator on this site does, is how the pattern is read from real data rather than assumed.
What is a reliability block diagram?
A drawing of a system as blocks for its items, connected in series where every item must work for the system to work and in parallel where any one of them is enough. The reliability or availability of the system is calculated from the blocks: series multiplies the probabilities, so the chain is weaker than its weakest link, and parallel redundancy raises it. IEC 61078 is the standard for the method, and it is the quickest way to see which item a plant's availability actually depends on.
Key terms
Plain-language definitions from our glossary for the concepts this article leans on.
Standards and further reading
- Reliability-centered maintenance (Nowlan and Heap, United Airlines, 1978) (US Department of Defense, DTIC)
- IEC 61649:2008 Weibull analysis (IEC)
- IEC 61078:2016 Reliability block diagrams (IEC)
- SAE JA1011 Evaluation criteria for reliability-centered maintenance processes (SAE International)
- ISO 14224:2016 Collection and exchange of reliability and maintenance data for equipment (ISO)
Related case studies and tools
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