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Current TARAM Process Details

Details of the Transport Airplane Risk Assessment Methodology (TARAM) process, including the constant failure rate (random) and the wear-out failure analysis based on the description in the Federal Aviation Administration’s (FAA’s) TARAM Handbook,¹ is included in this appendix.

From the three terms represented in Equations 2.1 and 2.2 in Chapter 2, the first two terms—the expected number of occurrences or events leading to fatalities or the corresponding rate per flight hour of such an event in the case of Equation 2.2, and the conditional probabilities (CPs)—depend on characteristics and failure mechanism of the item or equipment under consideration. The characteristics and likelihood of failure can vary over time. The hazard rate, or instantaneous failure rate, is useful to characterize the time-dependent failure rate by computing the conditional probability of failure in the next increment of time, conditional on survival up to time t.

The hazard rate function h(t) is a standard approach to model changes in reliability over time and is related to the survival function S(t) = 1–F(t), the probability that no failure occurred by time t, through the identity S(t) = e^{–∫t₀h(s)ds} (equivalently, for differentiable survival functions, it is written as h(t) = F’(t)/(1–F(t)), where F’ is the derivative of F). It follows that assuming a constant failure rate is equivalent to requiring that the time between failures is exponentially distributed. Other distributions for the time between failures will have time-dependent hazard functions.

The TARAM Handbook recognizes the need to distinguish between cases where the failure rate of the condition under study is either constant or increasing over time. These cases are discussed in the next two sections. The case where the rate is decreasing (also known as infant mortality) that refers to early failures, is not discussed separately in the TARAM Handbook. The reason given is that early failures are rare in transport-airplane continued operational safety (COS). If an early-failure issue is found, TARAM prescribes to assess the risk by following the wear-out guidance and worksheet. The TARAM Handbook suggests that the TARAM analysts contact the FAA Aircraft Certification Service, if necessary, for further guidance and information for the early-failure analysis.

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¹ Federal Aviation Administration, 2011, Transport Airplane Risk Assessment Methodology (TARAM) Handbook, PS-ANM-25-05, Washington, DC: Transport Airplane Directorate ANM-100.

TARAM’S CONSTANT FAILURE RATE (RANDOM) ANALYSIS

The constant failure rate analysis is performed when parts under study are considered to fail at random regardless of their age. In other words, the assumption is that the parts do not age (at least in reliability behavior). Under this assumption, the lifetime distribution is exponential.

The total uncorrected fleet risk defined as the expected number of events leading to fatalities given the condition under study is calculated as the product of the expected number of occurrences (failure events), the CP, and the severity of the unsafe outcomes (Equation 2.1). The expected number of occurrences during the remaining lifetime of the affected fleet and the CP depend on whether the condition under study has a constant failure rate or an increasing failure rate. Under the constant failure rate assumption, the expected number of such occurrences is calculated as the total number of flight hours remaining in the affected fleet (obtained from the determination of exposure factors in Figure 2.2) multiplied by the frequency of the occurrences under study or the rate of such occurrences per unit of time, as shown in Equation A.1.

The individual risk can be calculated from Equation 2.2, as the product of the rate of occurrence of the condition under study [i.e., F = (Rate of occurrence per flight hour)], the CP [i.e., P(Unsafe outcomes | occurrence)], and the Severity. Typically, the same failure rate as that for the total fleet risk is used in this calculation. Formulas for each of the five risks with the definition of all individual terms used for the risk calculations can be found in Table 2 in the TARAM Handbook.

As detailed later in Chapters 4 and 5, the uncertainties associated with the input parameter (failure rate) of the TARAM constant failure rate analysis need to be characterized. Lack of uncertainty characterization in the “rate of occurrence,” for example, could significantly influence the uncertainty in the total risk estimated from TARAM and that could mislead COS decision-making. (See Recommendation 7 in Chapter 5.)

TARAM’S WEAR-OUT FAILURE ANALYSIS

The wear-out analysis is performed when the parts under study are considered to be more likely to fail as they age. In this case, the failure rate, or hazard rate function, is an increasing function of time. Usual distributions used to model the failure time are Weibull and log-normal distributions.

When calculating the fleet risks under study, assuming a wear-out failure mode, the expected number of occurrences of the condition are calculated as the product of the expected number of airplanes that will fail owing to the condition under study (labeled as DA² in the handbook) and the probability that the condition (occurrence of the defect) is not detected before the unsafe outcome (or the non-detection probability, labeled as ND). To determine DA, it is necessary to determine the size of the affected fleet (obtained from the determination of exposure factors in Figure 2.2) and the failure rate distributions to calculate the probability of failure (owing to the condition under study) for each airplane during its remaining lifetime. The expected number of failures is calculated by taking a summation of the failure probability over the fleet of affected airplanes that have not yet failed.

Knowledge of the failure distribution is also used in the determination of the individual risk in the wear-out case, which is calculated as the product of the ND, the hazard function of the oldest plane at retirement, denoted as h_I, the CP, and the severity:

“ND * h_I” in Equation A.2 corresponds to F (rate of occurrence per flight hour) in Equation 2.2, representing the rate of occurrence of the condition under study. Again, the exact formulas for each of the five risks with the definition of all individual terms used for the risk calculations can be found in Table 2 in the TARAM Handbook.

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² Defined as “the expected number of airplanes that would experience the subject failure, if left undetected, during the time period under study,” in Chapter 5 in the TARAM Handbook.