**By Drew Troyer, Noria Corporation, December 2013, Reliableplant.com**

Increasingly, managers and engineers who are responsible for manufacturing and other industrial pursuits are incorporating a reliability focus into their strategic and tactical plans and initiatives. This trend is affecting numerous functional areas, including machine/system design and procurement, plant operations and plant maintenance. With its origins in the aviation industry, reliability engineering, as a discipline, has historically been focused primarily on assuring product reliability. More and more, these methods are being employed to assure the production reliability of manufacturing plants and equipment – often as an enabler to lean manufacturing. This article provides an introduction to the most relevant and practical of these methods for plant reliability engineering, including:

- Basic reliability calculations for failure rate, MTBF, availability, etc.
- An introduction to the exponential distribution – the cornerstone of the reliability methods.
- Identifying failure time dependencies using the versatile Weibull system.
- Developing an effective field data collection system.

*Introduction*

The origins of the field of reliability engineering, at least the demand for it, can be traced back to the point at which man began to depend upon machines for his livelihood. The Noria, for instance, is an ancient pump thought to be the world’s first sophisticated machine. Utilizing hydraulic energy from the flow of a river or stream, the Noria utilized buckets to transfer water to troughs, viaducts and other distribution devices to irrigate fields and provide water to communities. If the community Noria failed, the people who depended upon it for their supply of food were at risk. Survival has always been a great source of motivation for reliability and dependability.

While the origins of its demand are ancient, reliability engineering as a technical discipline truly flourished along with the growth of commercial aviation following World War II. It became rapidly apparent to managers of aviation industry companies that crashes are bad for business. Karen Bernowski, editor of *Quality Progress*, revealed in one of her editorials research into the media value of death by various means, which was conducted by MIT statistic professor Arnold Barnett and reported in 1994. Barnett evaluated the number of New York Times front-page news articles per 1,000 deaths by various means. He found that cancer-related deaths yielded 0.02 front-page news articles per 1,000 deaths, homicide yielded 1.7 per 1,000 deaths, AIDS yielded 2.3 per 1,000 deaths, and aviation-related accidents yielded a whopping 138.2 articles per 1,000 deaths!

The cost and high-profile nature of aviation related accidents helped to motivate the aviation industry to participate heavily in the development of the reliability engineering discipline. Likewise, due to the critical nature of military equipment in defense, reliability engineering techniques have long been employed to assure operational readiness. Many of our standards in the reliability engineering field are MIL Standards or have their origins in military activities.

Reliability engineering deals with the longevity and dependability of parts, products and systems. More poignantly, it is about controlling risk. Reliability engineering incorporates a wide variety of analytical techniques designed to help engineers understand the failure modes and patterns of these parts, products and systems. Traditionally, the reliability engineering field has focused upon product reliability and dependability assurance. In recent years, organizations that deploy machines and other physical assets in production settings have begun to deploy various reliability engineering principles for the purpose of production reliability and dependability assurance.

Increasingly, production organizations deploy reliability engineering techniques like Reliability-Centered Maintenance (RCM), including failure modes and effects (and criticality) analysis (FMEA,FMECA), root cause analysis (RCA), condition-based maintenance, improved work planning schemes, etc. These same organizations are beginning to adopt life cycle cost-based design and procurement strategies, change management schemes and other advanced tools and techniques in order to control the root causes of poor reliability. However, the adoption of the more quantitative aspects of reliability engineering by the production reliability assurance community has been slow. This is due in part to the perceived complexity of the techniques and in part due to the difficulty in obtaining useful data.

The quantitative aspects of reliability engineering may, on the surface, seem complicated and daunting. In reality, however, a relatively basic understanding of the most fundamental and widely applicable methods can enable the plant reliability engineer to gain a much clearer understanding about where problems are occurring, their nature and their impact on the production process – at least in the quantitative sense. Used properly, quantitative reliability engineering tools and methods enable the plant reliability engineering to more effectively apply the frameworks provided by RCM, RCA, etc., by eliminating some of the guesswork involved with their application otherwise. However, engineers must be particularly clever in their application of the methods because the operating context and environment of a production process incorporates more variables than the somewhat one-dimensional world of product reliability assurance due to the combined influence of design engineering, procurement, production/operations, maintenance, etc., and the difficulty in creating effective tests and experiments to model the multidimensional aspects of a typical production environment.

Despite the increased difficulty in applying quantitative reliability methods in the production environment, it is nonetheless worthwhile to gain a sound understanding of the tools and apply them where appropriate. Quantitative data helps to define the nature and magnitude of a problem/opportunity, which provides vision to the reliability in his or her application of other reliability engineering tools. This article will provide an introduction to the most basic reliability engineering methods that are applicable to the plant engineer that is interested in production reliability assurance. It presupposes a basic understanding of algebra, probability theory and univariate statistics based upon the Gaussian (normal) distribution (e.g. measure of central tendency, measures of dispersion and variability, confidence intervals, etc.).

It should be made clear that this paper is a brief introduction to reliability methods. It is by no means a comprehensive survey of reliability engineering methods, nor is it in any way new or unconventional. The methods described herein are routinely used by reliability engineers and are core knowledge concepts for those pursuing professional certification by the American Society for Quality (ASQ) as a reliability engineer (CRE). Several books on reliability engineering are listed in the bibliography of this article. The author of this article has found *Reliability Methods for Engineers* by K.S. Krishnamoorthi and *Reliability Statistics*by Robert Dovich to be particularly useful and user-friendly references on the subject of reliability engineering methods. Both are published by the ASQ Press.

Before discussing methods, you should familiarize yourself with reliability engineering nomenclature. For convenience, a highly abridged list of key terms and definitions is provided in the appendix of this article. For a more exhaustive definition of reliability terms and nomenclature, refer to MIL-STD-721 and other related standards. The definitions contained in the appendix are from MIL-STD-721.

*Basic mathematical concepts in reliability engineering*

Many mathematical concepts apply to reliability engineering, particularly from the areas of probability and statistics. Likewise, many mathematical distributions can be used for various purposes, including the Gaussian (normal) distribution, the log-normal distribution, the Rayleigh distribution, the exponential distribution, the Weibull distribution and a host of others. For the purpose of this brief introduction, we’ll limit our discussion to the exponential distribution and the Weibull distribution, the two most widely applied to reliability engineering. In the interest of brevity and simplicity, important mathematical concepts such as distribution goodness-of-fit and confidence intervals have been excluded.

**Failure rate and mean time between/to failure (MTBF/MTTF)**

The purpose for quantitative reliability measurements is to define the rate of failure relative to time and to model that failure rate in a mathematical distribution for the purpose of understanding the quantitative aspects of failure. The most basic building block is the failure rate, which is estimated using the following equation:

Where:

λ = Failure rate (sometimes referred to as the hazard rate)

*T *= Total running time/cycles/miles/etc. during an investigation period for both failed and non-failed items.

*r *= The total number of failures occurring during the investigation period.

For example, if five electric motors operate for a collective total time of 50 years with five functional failures during the period, the failure rate is 0.1 failures per year.

Another very basic concept is the mean time between/to failure (MTBF/MTTF). The only difference between MTBF and MTTF is that we employ MTBF when referring to items that are repaired when they fail. For items that are simply thrown away and replaced, we use the term MTTF. The computations are the same.

The basic calculation to estimate mean time between failure (MTBF) and mean time to failure (MTTF), both measures of central tendency, is simply the reciprocal of the failure rate function. It is calculated using the following equation.

Where:

θ = Mean time between/to failure

*T *= Total running time/cycles/miles/etc. during an investigation period for both failed and non-failed items.

*r *= The total number of failures occurring during the investigation period.

The MTBF for our industrial electric motor example is 10 years, which is the reciprocal of the failure rate for the motors. Incidentally, we would estimate MTBF for electric motors that are rebuilt upon failure. For smaller motors that are considered disposable, we would state the measure of central tendency as MTTF.

The failure rate is a basic component of many more complex reliability calculations. Depending upon the mechanical/electrical design, operating context, environment and/or maintenance effectiveness, a machine’s failure rate as a function of time may decline, remain constant, increase linearly or increase geometrically (Figure 1). The importance of failure rate vs. time will be discussed in more detail later.

*Figure 1. Different Failure Rates vs. Time Scenarios*

*The ‘Bathtub’ Curve*

Individuals that have received only basic training in probability and statistics are probably most familiar with the Gaussian or normal distribution, which is associated with familiar bell-shaped probability density curve. The Gaussian distribution is generally applicable to data sets where the two most common measures of central tendency, mean and median, are approximately equal. Surprisingly, despite the versatility of the Gaussian distribution in modeling probabilities for phenomenon ranging from standardized test scores to the birth weights of babies, it is not the dominant distribution employed in reliability engineering. The Gaussian distribution has its place in evaluating the failure characteristics of machines with a dominant failure mode, but the primary distribution employed in reliability engineering is the exponential distribution.

When evaluating the reliability and failure characteristics of a machine, we must begin with the much-maligned “bathtub” curve, which reflects the failure rate vs. time (Figure 2). In concept, the bathtub curve effectively demonstrates a machine’s three basic failure rate characteristics: declining, constant or increasing. Regrettably, the bathtub curve has been harshly criticized in the maintenance engineering literature because it fails to effectively model the characteristic failure rate for most machines in an industrial plant, which is generally true at the macro level. Most machines spend their lives in the early life, or infant mortality, and/or the constant failure rate regions of the bathtub curve. We rarely see systemic time-based failures in industrial machines. Despite its limitations in modeling the failure rates of typical industrial machines, the bathtub curve is a useful tool for explaining the basic concepts of reliability engineering.

*Figure 2. The Much-maligned ‘Bathtub’ Curve*

The human body is an excellent example of a system that follows the bathtub curve. People, and other organic species for that matter, tend to suffer a high failure rate (mortality) during their first years of life, particularly the first few years, but the rate decreases as the child grows older. Assuming a person reaches puberty and survives his or her teenage years, his or her mortality rate becomes fairly constant and remains there until age (time) dependent illnesses begin to increase the mortality rate (wearout). Numerous influences affect mortality rates, including prenatal care and mother’s nutrition, quality and availability of medical care, environment and nutrition, lifestyle choices and, of course, genetic predisposition. These factors can be metaphorically compared to factors that influence machine life. Design and procurement is analogous to genetic predisposition; installation and commissioning is analogous to prenatal care and mother’s nutrition; and lifestyle choices and availability of medical care is analogous to maintenance effectiveness and proactive control over operating conditions.

**The exponential distribution**

The exponential distribution, the most basic and widely used reliability prediction formula, models machines with the constant failure rate, or the flat section of the bathtub curve. Most industrial machines spend most of their lives in the constant failure rate, so it is widely applicable. Below is the basic equation for estimating the reliability of a machine that follows the exponential distribution, where the failure rate is constant as a function of time.

Where:

R(t) = Reliability estimate for a period of time, cycles, miles, etc. (t).

*e *= Base of the natural logarithms (2.718281828)

λ = Failure rate (1/MTBF, or 1/MTTF)

In our electric motor example, if you assume a constant failure rate the likelihood of running a motor for six years without a failure, or the projected reliability, is 55 percent. This is calculated as follows:

R(6) = 2.718281828-(0.1* 6)

R(6) = 0.5488 = ~ 55%

In other words, after six years, about 45% of the population of identical motors operating in an identical application can probabilistically be expected to fail. It is worth reiterating at this point that these calculations project the probability for a population. Any given individual from the population could fail on the first day of operation while another individual could last 30 years. That is the nature of probabilistic reliability projections.

A characteristic of the exponential distribution is the MTBF occurs at the point at which the calculated reliability is 36.78%, or the point at which 63.22% of the machines have already failed. In our motor example, after 10 years, 63.22% of the motors from a population of identical motors serving in identical applications can be expected to fail. In other words, the survival rate is 36.78% of the population.

We often speak of projected bearing life as the L10 life. This is the point in time at which 10% of a population of bearings should be expected to fail (90% survival rate). In reality, only a fraction of the bearings actually survive to the L10 point. We’ve come to accept that as the objective life for a bearing when perhaps we should set our sights on the L63.22 point, indicating that our bearings are lasting, on average, to projected MTBF – assuming, of course, that the bearings follow the exponential distribution. We’ll discuss that issue later in the Weibull analysis section of the article.

The probability density function (pdf), or life distribution, is a mathematical equation that approximates the failure frequency distribution. It is the pdf, or life frequency distribution, that yields the familiar bell-shaped curve in the Gaussian, or normal, distribution. Below is the pdf for the exponential distribution.

Where:

pdf(t) =Life frequency distribution for a given time (t)

*e *= Base of the natural logarithms (2.718281828)

λ = Failure rate (1/MTBF, or 1/MTTF)

In our electric motor example, the actual likelihood of failure at three years is calculated as follows:

*pdf(3) *= 01. * 2.718281828-(0.1* 3)

*pdf(3) *= 0.1 * 0.7408

*pdf(3) *= .07408 = ~ 7.4%

In our example, if we assume a constant failure rate, which follows the exponential distribution, the life distribution, or pdf for the industrial electric motors, is expressed in Figure 3. Don’t be confused by the declining nature of the pdf function. Yes, the failure rate is constant, but the pdf mathematically assumes failure without replacement, so the population from which failures can occur is continuously reducing – asymptotically approaching zero.

*Figure 3. The Probability Density Function (pdf)*

The cumulative distribution function (cdf) is simply the cumulative number of failures one might expect over a period of time. For the exponential distribution, the failure rate is constant, so the relative rate at which failed components are added to the cdf remains constant. However, as the population declines as a result of failure, the actual number of mathematically estimated failures decreases as a function of the declining population. Much like the pdf asymptotically approaches zero, the cdf asymptotically approaches one (Figure 4).

*Figure 4. Failure Rate and the Cumulative Distribution Function*

The declining failure rate portion of the bathtub curve, which is often called the infant mortality region, and the wear out region will be discussed in the following section addressing the versatile Weibull distribution.

*Weibull Distribution*

Originally developed by Wallodi Weibull, a Swedish mathematician, Weibull analysis is easily the most versatile distribution employed by reliability engineers. While it is called a distribution, it is actually a tool that enables the reliability engineer to first characterize the probability density function (failure frequency distribution) of a set of failure data to characterize the failures as early life, constant (exponential) or wear out (Gaussian or log normal) by plotting time to failure data on a special plotting paper with the log of the times/cycles/miles to failure plotted a log scaled X-axis versus the cumulative percent of the population represented by each failure on a log-log scaled Y-axis (Figure 5).

*Figure 5. The Simple Weibull Plot – Annotated*

Once plotted, the linear slope of the resultant curve is an important variable, called the shape parameter, represented by â, which is used to adjust the exponential distribution to fit a wide number of failure distributions. In general, if the â coefficient, or shape parameter, is less than 1.0, the distribution exhibits early life, or infant mortality failures. If the shape parameter exceeds about 3.5, the data are time dependent and indicate wearout failures. This data set typically assumes the Gaussian, or normal, distribution. As the â coefficient increases above ~ 3.5, the bell-shaped distribution tightens, exhibiting increasing kurtosis (peakedness at the top of the curve) and a smaller standard deviation. Many data sets will exhibit two or even three distinct regions. It is common for reliability engineers to plot, for example, one curve representing the shape parameter during run in and another curve to represent the constant or gradually increasing failure rate. In some instances, a third distinct linear slope emerges to identify a third shape, the wearout region. In these instances, the pdf of the failure data do in fact assume the familiar bathtub curve shape (Figure 6). Most mechanical equipment used in plants, however, exhibit an infant mortality region and a constant or gradually increasing failure rate region. It is rare to see a curve representing wearout emerge. The characteristic life, or η (lower case Greek “Eta”), is the Weibull approximation of the MTBF. It is always the function of time, miles or cycles where 63.21% of the units under evaluation have failed, which is the MTBF/MTTF for the exponential distribution.

*Figure 6. Depending upon the shape parameter, the Weibull failure density curve can assume several distributions, which is what makes it so versatile for reliability engineering.*

As a caveat to tie this tool back to excellence in maintenance and operations excellence, if we were to more effectively control the forcing functions that lead to mechanical failure in bearings, gears, etc., such as lubrication, contamination control, alignment, balance, appropriate operation, etc., more machines would actually reach their fatigue life. Machines that reach their fatigue life will exhibit the familiar wearout characteristic.

Using the β coefficient to adjust the failure rate equation as a function of time yields the following general equation:

Where:

*h(t) *= Failure rate (or hazard rate) for a given time (t)

*e *= Base of the natural logarithms (2.718281828)

θ = Estimated MTBF/MTTF

β = Weibull shape parameter from plot.

And, the following reliability function:

Where:

*R(t) *=Reliability estimate for a period of time, cycles, miles, etc. (t)

*e *= Base of the natural logarithms (2.718281828)

θ = Estimated MTBF/MTTF

β = Weibull shape parameter from plot.

And, the following probability density function (pdf):

Where:

*pdf(t) *=Probability density function estimate for a period of time,

cycles, miles, etc. (t)

*e *= Base of the natural logarithms (2.718281828)

θ = Estimated MTBF/MTTF

β = Weibull shape parameter from plot.

It should be noted that when the β equals 1.0, the Weibull distribution takes the form of the exponential distribution on which it is based.

To the uninitiated, the mathematics required to perform Weibull analysis may look daunting. But once you understand the mechanics of the formulas, the math is really quite simple. Moreover, software will do most of the work for us today, but it is important to have an understanding of the underlying theory so that the plant reliability engineer can effectively deploy the powerful Weibull analysis technique.

In our previously discussed example of electric motors, we previously assumed the exponential distribution. However, if Weibull analysis revealed early life failures by yielding a β shape parameter of 0.5, the estimate of reliability at six years time would be ~46%, not the ~55% estimated assuming the exponential distribution. In order to reduce wearout failures, we would need to lean on our suppliers to provide better-built and delivered quality and reliability, store the motors better to avoid rust, corrosion, fretting and other static wear mechanisms, and do a better job of installing and starting up new or rebuilt machines.

Conversely, if Weibull analysis revealed that the motors exhibited predominantly wearout-related failures, yielding a β shape parameter of 5.0, the estimate of reliability at six years time would be ~ 93%, instead of the ~55% estimated assuming the exponential distribution. For time-dependent wearout failures, we can perform scheduled overhaul or replacement assuming we have a good estimate of the MTBF/MTTF after we’ve reached the wearout region and a sufficiently small standard deviation so as to make high confidence rebuild/replace decisions that aren’t exceedingly costly.

In our motor example, assuming a β shape parameter of 5.0, the failure rate begins to rapidly increase after about five or six years, so we may want to edit our data to just focus upon the wearout region when estimating time-based replacement or rebuild time. Alternatively, we can improve the design, targeting the dominant failure mode(s) with the objective of decreasing “stress-strength” interferences. In other words, we can attempt to eliminate the machine’s frailties through design modification, the goal being to eliminate whatever is causing the time-dependent failures.

Assuming everything is constant, except the β shape parameter, Figure 7 illustrates the difference the β shape parameter has on the estimate of reliability assuming β shape values of 0.5 (early life), 1.0 (constant, or exponential) and 5.0 (wearout) for a range of time estimates. This graphic visually illustrates the concept of increasing risk vs. time (β = 0.5), constant risk vs. time (β = 1.0) and increasing risk vs. time (β = 5).

*Figure 7. Various Reliability Projections as a Function of Time for Different Weibull Shape Parameters*

*The Multi-Slope Weibull Plot*

Frequently, when drawing a best-fit regression line through the data points on a Weibull plot, the coefficient of correlation is poor, meaning the actual data points stray a great distance from regression line. This is assessed by examining the coefficient of correlation R, or more conservatively, R2, which denotes data variability. When correlation is poor, the reliability engineer should examine the data to evaluate if two or more patterns exist, which can denote major differences in failure modes, operating context, etc. Often, this produces two or more estimates of beta (Figure 8).

*Figure 8. An Example of a Multi-beta Weibull Plot*

As we see in our example in Figure 8, the data set works better when two distinct regression lines are drawn. The first line, exhibits a beta shape parameter of 0.5, suggesting early life failures. The second line exhibits a beta shape of 3.0, suggesting that the risk of failure increases as a function of time. It is common for complex equipment, particularly mechanical equipment, to experience “run-in” failures when new or recently rebuilt. As such, the risk of failure is highest just following initial start-up. Once the system works through its run-in period, which can take minutes, hours, days, weeks, months or years, depending upon the system type, the system enters a different risk pattern. In this example, the system enters a period where the risk of failure increases as a function of time once the system exits its run-in period.

The multi-beta offers the reliability engineer a more precise estimate of risk as a function of time. Armed with this knowledge, he or she is better positioned to take mitigating actions. For example, during the early life period, we’d be inclined to improve the precision with which we manufacture/rebuild, install and start-up. Moreover, we might add monitoring techniques and/or increase our monitoring frequency during the high risk period. Following the run-in period, we might introduce monitoring techniques that are targeted at the time-dependent wearout failures that are believed to affect the system, increase monitoring frequency accordingly or schedule “hard-time” preventive maintenance actions in some cases.

*Estimating System Reliability*

Once the reliability of components or machines has been established relative to the operating context and required mission time, plant engineers must assess the reliability of a system or process. Again, for the sake of brevity and simplicity, we’ll discuss system reliability estimates for series, parallel and shared-load redundant system (r/n systems).

**Series Systems**

Before discussing series systems, we should discuss reliability block diagrams. Not a complicated tool to use, reliability block diagrams simply map a process from start to finish. For a series system, Subsystem A is followed by Subsystem B and so forth. In the series system, the ability to employ Subsystem B depends upon the operating state of Subsystem A. If Subsystem A is not operating, the system is down regardless of the condition of Subsystem B (Figure 9).

To calculate the system reliability for a serial process, you only need to multiply the estimated reliability of Subsystem A at time (t) by the estimated reliability of Subsystem B at time (t). The basic equation for calculating the system reliability of a simple series system is:

Where:

*Rs(t) *– System reliability for given time (t)

*R1-n(t) *– Subsystem or sub-function reliability for given time (t)

So, for a simple system with three subsystems, or sub-functions, each having an estimated reliability of 0.90 (90%) at time (t), the system reliability is calculated as 0.90 X 0.90 X 0.90 = 0.729, or about 73%.

*Figure 9. Simple Serial System*

**Parallel Systems**

Often, design engineers will incorporate redundancy into critical machines. Reliability engineers call these parallel systems. These systems may be designed as active parallel systems or standby parallel systems. The block diagram for a simple two component parallel system is shown in Figure 10.

*Figure 10. Simple parallel system – the system reliability is increased to 99% due to the redundancy.*

To calculate the reliability of an active parallel system, where both machines are running, use the following simple equation:

Where:

*Rs(t) *– System reliability for given time (t)

*R1-n(t) *– Subsystem or sub-function reliability for given time (t)

The simple parallel system in our example with two components in parallel, each having a reliability of 0.90, has a total system reliability of 1 – (0.1 X 0.1) = 0.99. So, the system reliability was significantly improved.

There are some shortcut methods for calculating parallel system reliability when all subsystems have the same estimated reliability. More often, systems contain parallel and serial subcomponents as depicted in Figure 11. The calculation of standby systems requires knowledge about the reliability of the switching mechanism. In the interest of simplicity and brevity, this topic will be reserved for a future article.

*Figure 11. Combination System with Parallel and Serial Elements*

**r out of n Systems (r/n Systems)**

An important concept to plant reliability engineers is the concept of r/n systems. These systems require that r units from a total population in n be available for use. A great industrial example is coal pulverizers in an electric power generating plant. Often, the engineers design this function in the plant using an r/n approach. For instance, a unit has four pulverizers and the unit requires that three of the four be operable to run at the unit’s full load (see Figure 12).

*Figure 12. Simple r/n system example – Three of the four components are required.*

The reliability calculation for an r/n system can be reduced to a simple cumulative binomial distribution calculation, the formula for which is:

Where:

*Rs *= System reliability given the actual number of failures (r) is less than or equal the maximum allowable (k)

*r *= The actual number of failures

*k *= The maximum allowable number of failures

*n *= The total number of units in the system

*p *= The probability of survival, or the subcomponent reliability for a given time (t).

This equation is somewhat more complicated. In our pulverizer example, assuming a subcomponent reliability of 0.90, the equation works out as a summation of the following:

P(0) = 0.6561

P(1) = 0.2916

So, the likelihood of completing the mission time (t) is 0.9477 (0.6561 + 0.2916), or approximately 95%.

*Field Data Collection*

To employ the reliability analysis methods described herein, the engineer requires data. It is imperative to establish field data collection systems to support your reliability management initiatives. Likewise, as much as possible, you’ll want to employ common nomenclature and units so that your data can be parsed effectively for more detailed analysis. Collect the following information:

- Basic System Information
- Operating Context
- Environmental Context
- Failure Data

A good general system for data collection is described in the IEC standard 300-3-2. In addition to providing instructions for collecting field data, it provides a standard taxonomy of failure modes. Other taxonomies have been established, but the IEC standard represents a good starting point for your organization to define its own. Likewise, DOE standard NE-1004-92 offers a very nice standard nomenclature of failure causes.

An important benefit derived from your efforts to collect good field data is that it enables you to break the “random trap.” As I mentioned earlier, the bathtub curve has been much maligned – particularly in the Reliability-Centered Maintenance literature. While it’s true that Weibull analysis reveals that few complex mechanical systems exhibit time-dependent wearout failures, the reason, at least in part, is due to the fact that the reliability of complex systems is affected by a wide variety of failure modes and mechanisms. When these are lumped together, there is a “randomizing” effect, which makes the failures appear to lack any time dependency. However, if the failure modes were analyzed individually, the story would likely be very different (Figure 13). For certain, some failure modes would still be mathematically random, but many, and arguably most, would exhibit a time dependency. This kind of information would arm reliability engineers and managers with a powerful set of options for mitigating failure risk with a high degree of precision. Naturally, this ability depends upon the effective collection and subsequent analysis of field data.

*Figure 13. Good field data collection enables you to break the random trap.*

*Conclusions*

This brief introduction to reliability engineering methods is intended to expose the otherwise uninitiated plant engineer to the world of quantitative reliability engineering. The subject is quite broad, however, and I’ve only touched on the major reliability methods that I believe are most applicable to the plant engineer. I encourage you to further investigate the field of reliability engineering methods, concentrating on the following topics, among others:

- More detailed understanding of the Weibull distribution and its applications
- More detailed understanding of the exponential distribution and its applications
- The Gaussian distribution and its applications
- The log-normal distribution and its applications
- Confidence intervals (binomial, chi-square/Poisson, etc.)
- Beta distribution and its applications
- Bayesian applications of reliability engineering methods
- Stress-strength interference analysis
- Testing options and their applicability to plant reliability engineering
- Reliability growth strategies and management
- More detailed understanding of field data collection.

Most important, spend time learning how to apply reliability engineering methods to plant reliability problems. If your interest in reliability engineering methods is high, I encourage you to pursue professional certification by the American Society for Quality as a reliability engineer (CRE).

**References**

Troyer, D. (2006) Strategic Plant Reliability Management Course Book, Noria Publishing, Tulsa, Oklahoma.

Bernowski, K (1997) “Safety in the Skies,” *Quality Progress*, January.

Dovich, R. (1990) Reliability Statistics, ASQ Quality Press, Milwaukee, WI.

Krishnamoorthi, K.S. (1992) Reliability Methods for Engineers, *ASQ Quality Press*, Milwaukee, WI.

MIL Standard 721

IEC Standard 300-3-3

DOE Standard NE-1004-92

**Appendix: Select reliability engineering terms from MIL STD 721**

**Availability** – A measure of the degree to which an item is in the operable and committable state at the start of the mission, when the mission is called for at an unknown state.

**Capability** – A measure of the ability of an item to achieve mission objectives given the conditions during the mission.

**Dependability** – A measure of the degree to which an item is operable and capable of performing its required function at any (random) time during a specified mission profile, given the availability at the start of the mission.

**Failure** – The event, or inoperable state, in which an item, or part of an item, does not, or would not, perform as previously specified.

**Failure, dependent** – Failure which is caused by the failure of an associated item(s). Not independent.

**Failure, independent** – Failure which occurs without being caused by the failure of any other item. Not dependent.

**Failure mechanism** – The physical, chemical, electrical, thermal or other process which results in failure.

**Failure mode** – The consequence of the mechanism through which the failure occurs, i.e. short, open, fracture, excessive wear.

**Failure, random** – Failure whose occurrence is predictable only in the probabilistic or statistical sense. This applies to all distributions.

**Failure rate** – The total number of failures within an item population, divided by the total number of life units expended by that population, during a particular measurement interval under stated conditions.

**Maintainability** – The measure of the ability of an item to be retained or restored to specified condition when maintenance is performed by personnel having specified skill levels, using prescribed procedures and resources, at each prescribed level of maintenance and repair.

**Maintenance, corrective** – All actions performed, as a result of failure, to restore an item to a specified condition. Corrective maintenance can include any or all of the following steps: localization, isolation, disassembly, interchange, reassembly, alignment and checkout.

**Maintenance, preventive** – All actions performed in an attempt to retain an item in a specified condition by providing systematic inspection, detection and prevention of incipient failures.

**Mean time between failure (MTBF)** – A basic measure of reliability for repairable items: the mean number of life units during which all parts of the item perform within their specified limits, during a particular measurement interval under stated conditions.

**Mean time to failure (MTTF)** – A basic measure of reliability for non-repairable items: The mean number of life units during which all parts of the item perform within their specified limits, during a particular measurement interval under stated conditions.

**Mean time to repair (MTTR)** – A basic measure of maintainability: the sum of corrective maintenance times at any specified level of repair, divided by the total number of failures within an item repaired at that level, during a particular interval under stated conditions.

**Mission reliability** – The ability of an item to perform its required functions for the duration of specified mission profile.

**Reliability** – (1) The duration or probability of failure-free performance under stated conditions. (2) The probability that an item can perform its intended function for a specified interval under stated conditions. For non-redundant items this is the equivalent to definition (1). For redundant items, this is the definition of mission reliability.

**About the author:**

Drew D. Troyer is a champion of effective reliability management and passionate about helping companies find hidden profits inside their plants. As a highly sought consultant to Fortune 500 manufacturing firms, award-winning columnist and teacher, he understands both management expectations and plant-floor realities. Troyer is a Certified Reliability Engineer (CRE), a Certified Maintenance and Reliability Professional (CMRP), and chairs the standards committee of the Society for Maintenance and Reliability Professionals (SMRP). Contact Drew at 800-597-5460.