LIFE-CYCLE
4.2 TIME-VARIANT PERFORMANCE INDICATORS
4.2.1 Time-Variant Reliability
In structural reliability theory, the safe condition is the one in which the failure of the investigated component / system does not occur. For a structural component with resistance r and load effect s, its performance function is:
s r
g (4.1) The probability that this component fails is:
] 0
) [
( gP
Pf component
(4.2)
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For a structural system with at least two failure modes, its failure probability is defined as the probability of violating any of the limit states that are defined by its failure modes: where gi = system performance function with respect to failure mode i. Due to the usual assumption of Gaussian distribution of performance functions, the reliability index associated with the evaluated structural component / system is given by:
)
1( Pf
)
E (4.4)
where ) = standard normal cumulative distribution function.
In many previous studies, loads and resistances are considered as time-independent random variables (Hendawi and Frangopol 1994, Wang and Wen 2000, Imai and Frangopol 2001, Zhao and Ono 2001, Lin and Frangopol 1996). Accordingly, the probability of failure obtained from Equation (4.3) is kept unchanged during the lifetime of a structure. However, in most practical cases, resistances and loads of a structure vary with time. In general, the resistances deteriorate and loads increase over time. Therefore, Equations (4.1) to (4.4) considering the time effects can be rewritten as:
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where t = time. It should be noted that the probabilities of failure obtained from Equations (4.6) and (4.7) are instantaneous. They define the probability of failure at a point-in-time, rather than evaluate the probability within a specified time interval, which is known as cumulative probability of failure. Since the instantaneous probability of failure changes over time, it is time-variant. In this chapter, the “time-variant” performance indicators (reliability, redundancy, and risk) are considered as
“point-in-time”.
4.2.2 Time-Variant Redundancy
System redundancy has been defined as the availability of system warning before the occurrence of structural collapse (Okasha and Frangopol 2009a). Several studies have been performed in presenting measures of quantifying redundancy for structural design or assessment (Frangopol and Curley 1987, Okasha and Frangopol 2009a, Blagojevic and Ziha 2008, Ghosn and Moses 1998, Liu et al. 2001, Frangopol 2011).
However, no agreement has been reached on redundancy measures yet. In this chapter,
the time-variant redundancy index provided in (Frangopol 2011) is used:
( ) s( ) fc( )
RI t E t E t (4.9) where Es( )t = system reliability index at time t, and Efc( )t = reliability index associated with the probability of the first component failure at time t.
The larger the difference between these two reliability indices, the higher redundancy the system has. This difference can be interpreted as the availability of system warning before failure. However, the redundancy defined in Equation (4.9) cannot be used as the only metric for assessing structural safety. For example, consider
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two structures whose system reliability indices are 5.0 and 3.0, respectively, and the reliability indices of first component failure are 3.0 and 1.0, respectively. Obviously, the two systems have the same redundancy index (i.e. 2.0); however, the first system is much more reliable than the second one. Therefore, the redundancy index defined in Equation (4.9) should be combined with the information on other performance indicators (such as system reliability and risk) to obtain a more complete assessment of time-variant structural performance.
4.2.3 Time-Variant Risk
Risk has become an increasingly important performance indicator. It is defined as the combined effect of probabilities and consequences of some failure or disaster in a given context:
( ) f( ) ( )
R t P t uC t (4.10) where R(t) = risk caused by a failure in a given context at time t, Pf(t) = probability of failure at time t, and C(t) = consequences caused by the failure at time t. The time-variant probability of a component or system failure, Pf(t), can be obtained after identifying the performance function of the component or the failure modes of the system.
Consequences caused by the failure of components or system consist of two parts:
(a) direct consequences, CDIR(t), which are related to local components failure; and (b) indirect consequences, CIND(t), which are associated with subsequent system failure (Baker et al. 2006). Direct consequences are considered proportional to the initial damage since they include only the commercial loss aspect (i.e., the cost required to
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replace the damaged component / system), while the indirect consequences are not proportional to the initial damage because they consist of several loss aspects, such as safety loss, commercial loss and environmental loss (Hessami 1999). An event-tree model for a general case where component i fails is shown in Figure 4.1. In this figure, Fcomp,i denotes the event that component i fails; Fsubsys|Fcomp i, and Fsubsys|Fcomp i,
represent the events that the subsequent system fails and survives given the failure of component i, respectively. The subsequent system discussed herein is the system without component i. In branch b1 , only direct consequence exists since only
component i fails and the subsequent system survives. However, in branch b2, both direct and indirect consequences occur because the subsequent system fails after failure of component i.
Based on the classification of the consequences (i.e., direct and indirect consequences), the risk at time t caused by the failure of component i can be divided into direct risk RDIR,i(t) and indirect risk RIND,i(t), which are computed as:
,( ) , ,( ) ,( )
DIR i f comp i DIR i
R t P t uC t (4.11)
,( ) , ,( ) , | ,( ) ,( )
IND i f comp i f subsys comp i IND i
R t P t uP t uC t (4.12) where Pf comp i, ,( )t = failure probability of component i at time t, Pf subsys comp i, | ,( )t = probability of subsequent system failure at time t given the failure of component i,
,( )
DIR i
C t = direct consequences at time t associated with the failure of component i (i.e., the cost to replace this component), and CIND i,( )t = indirect consequences at time t caused by the failure of component i (i.e., the cost to rebuild the subsequent system,
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safety loss, and environmental loss). Finally, the total risk caused by the failure of component i is:
,( ) ,( ) ,( )
TOT i DIR i IND i
R t R t R t (4.13)