PSI - Issue 19

Aaron Stenta et al. / Procedia Structural Integrity 19 (2019) 27–40 Stenta and Panzarella / Structural Integrity Procedia 00 (2019) 000–000

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some sort of decision needs to be made by selecting one out of a number of possible options. The blue steps are action items. The red steps indicate when costs C are subtracted from H and the green steps are when benefits B are added to H . The average annual return Life h H t  , obtained by dividing H by the total lifetime Life t , is actually what needs to be optimized by finding the optimal set of parameters based on all current and expected knowledge about the system.     ( ) H B C   x x x (1) The lifetime operation of a piece of equipment can be thought of as being broken up into a series of cycles of operation, inspection, and repair. If the conditions of each cycle are chosen properly, then the annual return will be maximized. If a component is operated for some period of time, some amount of benefit is derived from that operation. At the same time, the component is being damaged in some way due to that operation, and this puts the component at risk of failure. The risk of failure depends on its probability as well as its cost, should it happen. For example, if installation, inspection, repairs, and the time value of money are all ignored for the moment, a simpler situation can lead to some insight about when the next inspections should be scheduled (when the next cycle should begin) in order to maximize the return. Consider the expected return defined in Eq. (2), written simply in terms of the benefit and expected cost of the next failure, based upon its probability of occurrence. Both the benefit and the probability of failure are increasing functions of the operation time, but they increase at different rates. The expected cost is   F F P t C  , and the expected return is       F F H t B t P t C    (2) Typically, the average benefit will increase nearly linearly with operating time,   F B t t    , since product is produced and sold at some nearly constant monthly or yearly rate. The probability of failure is normally very small to begin with but then increases super-linearly at a later time as illustrated in Figure 2a. Because of this, one would expect   F F P t C  to be initially less than the benefit curve   B t but then rapidly approach and intersect it at some later time, at which point 0 H  . Since H is initially zero, then becomes positive and returns to zero at some later time, there must be some intermediate time where H reaches a maximum positive value. The time at which this happens is the optimal operating time without any inspections or repair, because it maximizes the return H . This optimal time occurs when the distance between the benefit curve and the cost curve is greatest, as illustrated in Figure 2b, assuming $50 F C  million, $2 B   million/year and the   F P t shown in Figure 2a. Geometrically, this is also the point where the tangent line to the expected cost curve is parallel to the benefit curve.

Fig. 2. (a) Typical Probability Distribution Function (PDF) and Cumulative Distribution Function (CDF) for a random variable such as the failure time that has a mean of 5 years and a standard deviation of 1 year; (b) If one chooses to operate over some period of time, there is some probability of failure P F with cost C F , but a benefit B is also achieved. The mean expected return is then H=B-P F ∙C F . Increasing the operating time increases the benefit but also increases the probability of failure.