The Bathtub Curve

Frequently, tlie failure rate of equipment exliibits tliree stages a break-in stage with a declining failure rate, a useful life stage cliaracterized by a fairly constant 

In tlie cases of the batlitub curve, failure rate during useful life is constant. Letting tliis constant be a and substituting it for Z(t) in Eq. (20.3.18) yields 

The Weibull distribution provides a iiiatliematical model of all tliree stages of the batlitub curve. Tliis is now discussed. An assumption about failure rate tliat reflects all tliree stages of tlie batlitub curve is 

Equation (20.4.3) defines tlie pdf of the Weibull distribution. Tlie exponential distribution, whose pdf is given in Eq. (20.4.1), is a special case of the Weibull distribution witli p = 1. Tlie variety of assumptions about failure rate and tlie probability distribution of time to failure tliat can be accommodated by the Weibull distribution make it especially attractive in describing failure time distributions in industrial and process plant applications. 

Estimates of the panuiieters a and p in Eq. (20.4.3) can be obtained by using a grapliical procedure described by Bury. Tliis procedure is based on tlie fact tliat 

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Monitoring the trends of a machine-train or process system will provide the ability to prevent most catastrophic failures. The trend is similar to the bathtub curve used to schedule preventive maintenance. The difference between the preventive and predictive bathtub curve is that the latter is based on the actual condition of the equipment, not a statistical average. 

Early failures and wear out failures are reflected in the curve known as the bathtub curve (see Figure 11.14). 

The Weibull distribution provides a mathematical model of all three stages of the bathtub curve. An assumption about failure rate that reflects all three stages of the bathtub stage is... 

With shape parameter / < 1 a decreasing failure rate can be simulated, insuch as on the left side of the bathtub curve (Fig. 5.9.6). For / =1 the Weibull distribution reduces to a pure exponential with a constant failure rate that is given by the reciprocal of the characteristic time T ... 

The lifetime of a population of units at the component, board, box, or system level can be divided into three distinct periods. This is most often defined by the so-called reliability bathtub curve (Fig. 6.16). The bathtub curve describes the cradle-to-grave failure rates or frequency of failures as a function of time. The curve is divided into three distinct areas early failure rate (also known as infant mortality), the useful life period, and the wearout failure period. The infant-mortality portion of the curve, also known as the early life period, is the initial steep slope from the start to... 

The graphical results of various Ss on the Weibull analysis can be seen in Fig. 6.17. If all curves are taken together, the result is the bathtub curve. 

Figure 6.17 Weibull plots and the bathtub curve (a) infant-mortality period, (b) infant mortality and useful life period and (c) useful life and end-of-life period.

The bathtub curve represents the curve of the failure rate of a product, which is denoted by X(t) (L. Scheidt, et al. 1994. Niu Peng-zhi, et al. 2007. Wang Rong-hua, et al. 2002) In terms of the maintainability of a product, the failure rate points to the malfunction rate. The rate of failure is also called the rate of damage failure. The rate of safety damage shows the damage probability of system, which is the same as the rate of failure. RD (t) is thus equal to X(t). 

An item s failure rate is generally not a single value— it will vary with time and the age of the item. The bathtub curve, shown in Figure 16.9, illustrates this phenomenon (the term bathtub comes from the rather fanciful resemblance of the shape of the overall failure rate to that of a bathtub). 

This was based on the bathtub curve (Fig. 1) which depicts the increase in the failure rate of items in time. Time-based maintenance does not account for the fact that the condition of the items depends not only on elapsed time but also on operational and environmental conditions. Based on advances in diagnostic techniques in the 1970s, condition-based maintenance (or predictive maintenance) was introduced. Here, maintenance activities occur when symptoms of wear or failure are determined either by monitoring or diagnosis, i.e., maintenance effectiveness depends on the existence of appropriate diagnostic techniques. 

The shape of the plot in Figure 3-2 is characteristic of many components and well known to reliability engineers. The shape is called the "bathtub curve." Three regions are distinct. In the early portion of the plot, failure rates are higher. This area is called "infant mortality." The middle portion of the curve is known as "useful Ufe." The final portion of the curve is called "end of tife" or "wearout region."... 

Failures occur when some "stress" exceeds the corresponding "susceptibility" in any component. As a concept it makes perfect sense in the context of mechanical engineering. The stress is usually a force, and the susceptibility is the point where the mechanical component can no longer resist that force. An analogous concept has been developed for electronic components (Ref. 3). Simulations using the stress-susceptibility concept generate failure rate curves similar to the bathtub curve (Ref. 4 and 5). 

The end of the bathtub curve occurs when the strength of the product declines (susceptibility increases). This is commonly known as "wearout." Wearout occurs after several years but the mechanisms vary considerably depending on the type of component. 

Field data may include failures occurring during the wearout portion of the bathtub curve... 

As can be seen in Eq. (9.23) the survival probability of a component is completely determined by its failure rate )t(t). Its general shape is known as the bathtub curve , which is similar to the curve of human mortality. The bathtub curve is shown in Fig. 9.12. 

The shape factor a, which is the density, distribution and failure rate function, and represents specific regularity failure occurrence, in function with the Peta parameter and acquires three different shapes P <1, P = 1, P> 1, which defines the "bathtub curve" as shown in figure 4. 

Bertholon model can be generalized into a model to 7 parameters characterizing the three phases of the bathtub curve a first Weibull law with P < I for the phase of youth failure, an exponential law for the phase of occasional failures and a second Weibull with P > for the wear phase. It corresponds to three blocks in series, the first is a Weibull, initiated at t = 0 (y = 0) and limited to duration T, and the other two corresponding to the Bertholon model. The occurrence of failure can be simulated by the formula 4 under Excel. 

Update OHAs should be scheduled to address the bathtub curve for the end product. The bathtub curve predicts that accident and incident rates tend... 

Statistical values can also be used to determine expected periods of optimum performance in the life cycle of products, systems, hardware, or equipment. For example, if the life cycle of humans were plotted on a curve, the period of their lives that may be considered most useful, in terms of productivity and success, could be represented as shown in Figure 5.4. This plotted curve is often referred to as the bathtub curve because of its obvious shape. A similar curve can be used to determine the most productive period of a product s life cycle according to the five known phases of that life cycle, as discussed in Chapter 3. The resultant curve, known as a product s reliability curvey would resemble the curve that appears in Figure 5.5. During the breakin period, failures in the system may occur more frequently, but decreasingly less frequently as the curve begins to level toward the useful life period. Then, as the system reaches the end of its useful life and approaches wearout, more frequent failure experience is likely until disposal. 

As proof-test intervals are increased, it is also important to recognize that the useful life of the equipment as it relates to the random failure rate portion of the bathtub curve should not be exceeded. 

One long established concept in risk engineering is the bathtub curve. See Fig. 7.1. 

The bathtub curve is a longstanding concept and describes a graph of maintenance levels versus maintenance costs (see Fig. 7.1). These costs normally gradualfy decrease as equipment is bedded in, stay steady for the projected lifetime of equipment and then start to show a distinct climb again as equipment reaches the end of its useful life. 

The lifetime of an entire population of products often is graphically represented by a set of curves collectively called the bathtub curve. Bathtub curve has been depicted in Fig. VII/1.2.2-1. The bathtub curve consists of three periods. First is an infant mortality or burnt-in period with a decreasing failure rate showing early-life failure. These... 

D.J. Wilkins, The Bathtub Curve and Product Failure Behavior Part One - the Bathtub Curve, Infant Mortality and Burn-in, ReliaSoff Reliability Field Consultant, November 2002. Reliability hotwire e magazine issue 21. 

The management of aging of safety-related equipment has to, in particular, address carefully the approach to wear-out in the bathtub curve (Fig. 6.1). Management indicators - key performance indicators or KPIs - are essential tools for this [1,2],... 

In doing so, the failure rate X plays an important role, several roles respectively, and it a priori assumes pure random failures. For the scope of the standard, which is safety related parts of control systems , this is a very simple comparison standard. It fits quite well, if a constant failure rate X can be suggested, e g. for electronical products, which can be connected to the horizontal part of the bathtub curve . In the VDW s study on operational dependability, by contrast, the Weibull distribution is used, because at machine tools the failures are not purely random, but the failure rate rises as the operating hours increase (as a consequence of wear and tear, ageing). 

This fact led to the definition of the bathtub curve, which is often used as a reference model in statistics. It facilitates the observation and its extent and sufficiently offsets variances especially for electric components (Fig. 3.1). 

The bathtub curve shows three areas over time. The early failure phase describes the time frame in which the failure behavior is not sufficiently developed through unknown influences, environment parameters, correct materials and bias points. This should be investigated for the development of components within the context of the design verification so that phase 2, the usage phase, can be entered at the beginning of series production. The usage phase should be designed in a way that the failure rate only starts after the expiration of the statistical life expectation of the components. In reality the failure rate is placed below the bathtub curve, as far as necessary so that an age induced increase can be seen and a sufficient robustness level ensures that the statistical life expectation is achieved. ISO 26262 does not mention any requirements, for example in order to prevent early failure behavior. 

However, particularly for electric components proven handbook data are used as reference, since the bathtub curve and the material dependency in such formulas... 

In the automobile industry, AEC (Q) 100 is used for complex components. It is a standard for the qualification of electric components. Simple components as resistors or capacitors are not covered in this standard. Since these simple components would often push aU statistic boimdaries through their variety of elements, such statistic observations are often insufficient for safety engineering. The risk for such simple components is that harmful components can be delivered to the production undetected. This is why the eligibility and whether the components are actually sufficiently dimensioned for their case of application are tested in the context of the qualification of the entire electric assembly group. The value for failiue rates is taken from the reliability handbooks. However, for the correct qualification including the proof of lifetime efficiency of the entire electronic assembly group it is assumed that the simple components is within the constant phase of failure rates of the bathtub curve. 

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