to bias. Methods are available to reduce this bias; however, they increase number of events which
must be analyzed and the time required by the expert panel. Guidance concerning the use of
expert elicitation should be obtained in consultation with CECW-ED.
D-4. Analysis Procedures. Guidance for conducting reliability analyses using hazard functions,
historical frequencies of occurrence and expert elicitation are available from various sources in the
literature and upon consultation with CECW-ED. The reliability index provides a means to
express reliability as a function of the means and standard deviations of C and D, where C and D
are functions expressing the capacity and demand associated with the performance mode. A
reliability index based analysis for a typical mode of performance should be conducted in
accordance with ETL 1110-2-532.
D-5. Overall System Reliability. Reliability for a number of components or a number of modes of
performance, may be used to estimate the overall reliability of a structure. To gain insight into
this approach, consider two extreme cases, the series system and the parallel system.
a. Series System. In a series system, the system will perform unsatisfactorily if any one
component performs unsatisfactorily. If a system has n components in series, the probability of
reliability of the system, or probability that all components will perform satisfactorily, is the
product of the component reliabilities:
R = R1R2R3....Rn = (1-p1)(1-p2)(1-p3)...(1-pn)
b. Simple Parallel System. In a parallel system, the system will only perform
unsatisfactorily if all components perform unsatisfactorily. Thus, the reliability is unity minus the
probability that all components perform unsatisfactorily, or:
R = 1 - p1p2p3....pn
c. Parallel and Series Systems. Solutions are available for systems requiring r-out-of-n
operable components, which may be applicable to problems such as dewatering with multiple
pumps, or closing a gate bay with emergency bulkheads. Subsystems involving independent
parallel and series systems can be mathematically combined by standard techniques.
(1) Upper and lower bounds on system reliability can be determined by considering all
components to form parallel and series systems, respectively; however, the resulting bounds may
be so broad as to be unpractical. A number of procedures are found in the references to narrow
the bounds.
(2) Civil engineering systems such as locks and dams (or even building frames) are
complex and may have many performance modes. Some of these may not be independent; for
instance several performance modes may be correlated to the occurrence of a high or low pool
level. Earth pressures, sliding, and overturning performance are all correlated to shear strength.
Rational estimation of the overall reliability of a lock and dam is a topic that is undergoing further
research.
d. A Practical Approach. In many systems, the reliability of a few subsystems or
components may govern the reliability of the entire system; navigation systems and hydropower
plants are no exception. Thus, developing a means to characterize and compare the reliability of
these components as a function of time are sufficient to make engineering judgements to aid in
prioritizing O&M expenditures.
D-4
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